symmetry log of sum of exp

Percentage Accurate: 53.6% → 98.6%

Time: 12.2s

Alternatives: 18

Speedup: 1.5×

• could not determine a ground truth

  1. a = 1.5539780270441005e-61
  2. b = 1.6721346955172488e+22

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: 53.6% 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: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(1.0 + exp(a))
	return Float64(Float64(b * fma(Float64(b * -0.5), (t_0 ^ -2.0), Float64(fma(b, 0.5, 1.0) / t_0))) + log1p(exp(a)))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]}, N[(N[(b * N[(N[(b * -0.5), $MachinePrecision] * N[Power[t$95$0, -2.0], $MachinePrecision] + N[(N[(b * 0.5 + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

5. Simplified 72.5%

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

7. Applied egg-rr 72.5%

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

Alternative 2: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} + e^{b} \leq 1.1:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (+ (exp a) (exp b)) 1.1) (/ b (+ 1.0 (exp a))) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if ((exp(a) + exp(b)) <= 1.1) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(a) + Math.exp(b)) <= 1.1) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if (math.exp(a) + math.exp(b)) <= 1.1:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) + exp(b)) <= 1.1)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision], 1.1], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 a) (exp.f64 b)) < 1.1000000000000001

   1. Initial program 9.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 50.0%

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

   7. Applied egg-rr 50.0%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \mathsf{log1p}\left(e^{a}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 50.2

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

   10. Simplified 50.2%

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

   11. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   12. 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 49.0

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

   13. Simplified 49.0%

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

   if 1.1000000000000001 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 97.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 95.1

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

   5. Simplified 95.1%

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

Alternative 3: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} + e^{b} \leq 1.5:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(b, -0.25, 0.5\right), \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.125, 0.5\right), \log 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (+ (exp a) (exp b)) 1.5)
   (/ b (+ 1.0 (exp a)))
   (fma a (fma b -0.25 0.5) (fma b (fma b 0.125 0.5) (log 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if ((exp(a) + exp(b)) <= 1.5) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = fma(a, fma(b, -0.25, 0.5), fma(b, fma(b, 0.125, 0.5), log(2.0)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) + exp(b)) <= 1.5)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = fma(a, fma(b, -0.25, 0.5), fma(b, fma(b, 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[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision], 1.5], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(b * -0.25 + 0.5), $MachinePrecision] + N[(b * N[(b * 0.125 + 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 49.8%

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

   7. Applied egg-rr 49.8%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \mathsf{log1p}\left(e^{a}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 49.9

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

   10. Simplified 49.9%

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

   11. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   12. 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 48.6

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

   13. Simplified 48.6%

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

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

   1. Initial program 97.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 96.7%

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

   6. Taylor expanded in a around 0

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

      1. distribute-rgt-out N/A

         \[\leadsto \log 2 + \left(a \cdot \left(\frac{1}{2} + b \cdot \left(\color{blue}{b \cdot \left(\frac{-1}{8} + \frac{1}{8}\right)} - \frac{1}{4}\right)\right) + b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \log 2 + \left(a \cdot \left(\frac{1}{2} + b \cdot \left(b \cdot \color{blue}{0} - \frac{1}{4}\right)\right) + b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right) \]

      3. mul0-rgt N/A

         \[\leadsto \log 2 + \left(a \cdot \left(\frac{1}{2} + b \cdot \left(\color{blue}{0} - \frac{1}{4}\right)\right) + b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \log 2 + \left(a \cdot \left(\frac{1}{2} + b \cdot \color{blue}{\frac{-1}{4}}\right) + b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \log 2 + \left(a \cdot \left(\frac{1}{2} + b \cdot \color{blue}{\left(0 - \frac{1}{4}\right)}\right) + b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right) \]

      6. mul0-rgt N/A

         \[\leadsto \log 2 + \left(a \cdot \left(\frac{1}{2} + b \cdot \left(\color{blue}{b \cdot 0} - \frac{1}{4}\right)\right) + b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \log 2 + \left(a \cdot \left(\frac{1}{2} + b \cdot \left(b \cdot \color{blue}{\left(\frac{-1}{8} + \frac{1}{8}\right)} - \frac{1}{4}\right)\right) + b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \log 2 + \left(a \cdot \left(\frac{1}{2} + b \cdot \left(\color{blue}{\left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)} - \frac{1}{4}\right)\right) + b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(\frac{1}{2} + b \cdot \left(\left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right) - \frac{1}{4}\right)\right) + b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right) + \log 2} \]

      10. associate-+l+ N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{1}{2} + b \cdot \left(\left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right) - \frac{1}{4}\right)\right) + \left(b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right) + \log 2\right)} \]

   8. Simplified 95.3%

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

Alternative 4: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} + e^{b} \leq 1.1:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.125, 0.5\right), \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (+ (exp a) (exp b)) 1.1)
   (/ b (+ 1.0 (exp a)))
   (fma b (fma b 0.125 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if ((exp(a) + exp(b)) <= 1.1) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = fma(b, fma(b, 0.125, 0.5), log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) + exp(b)) <= 1.1)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = fma(b, fma(b, 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[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision], 1.1], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 a) (exp.f64 b)) < 1.1000000000000001

   1. Initial program 9.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 50.0%

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

   7. Applied egg-rr 50.0%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \mathsf{log1p}\left(e^{a}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 50.2

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

   10. Simplified 50.2%

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

   11. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   12. 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 49.0

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

   13. Simplified 49.0%

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

   if 1.1000000000000001 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 97.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 96.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right) + \log 2} \]

      2. distribute-rgt-out N/A

         \[\leadsto b \cdot \left(\frac{1}{2} + \color{blue}{b \cdot \left(\frac{-1}{8} + \frac{1}{4}\right)}\right) + \log 2 \]

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 93.6

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

   8. Simplified 93.6%

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

Alternative 5: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} + e^{b} \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.125, 0.5\right), \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (+ (exp a) (exp b)) 1.1)
   (fma (* b b) (* b -0.16666666666666666) b)
   (fma b (fma b 0.125 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if ((exp(a) + exp(b)) <= 1.1) {
		tmp = fma((b * b), (b * -0.16666666666666666), b);
	} else {
		tmp = fma(b, fma(b, 0.125, 0.5), log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) + exp(b)) <= 1.1)
		tmp = fma(Float64(b * b), Float64(b * -0.16666666666666666), b);
	else
		tmp = fma(b, fma(b, 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[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision], 1.1], N[(N[(b * b), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision] + b), $MachinePrecision], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 a) (exp.f64 b)) < 1.1000000000000001

   1. Initial program 9.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-exp.f64 5.5

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

   5. Simplified 5.5%

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

   6. Taylor expanded in b around inf

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

      1. distribute-lft-in N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. rgt-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 5.2

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

   8. Simplified 5.2%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       9. lower-*.f64 47.8

          \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot -0.16666666666666666}, b\right) \]

   11. Simplified 47.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)} \]

   if 1.1000000000000001 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 97.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 96.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right) + \log 2} \]

      2. distribute-rgt-out N/A

         \[\leadsto b \cdot \left(\frac{1}{2} + \color{blue}{b \cdot \left(\frac{-1}{8} + \frac{1}{4}\right)}\right) + \log 2 \]

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 93.6

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

   8. Simplified 93.6%

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

Alternative 6: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} + e^{b} \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.125, 0.5\right), \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (+ (exp a) (exp b)) 1.1)
   (fma (* b b) (* b -0.16666666666666666) b)
   (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) + exp(b)) <= 1.1) {
		tmp = fma((b * b), (b * -0.16666666666666666), b);
	} 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 (Float64(exp(a) + exp(b)) <= 1.1)
		tmp = fma(Float64(b * b), Float64(b * -0.16666666666666666), b);
	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[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision], 1.1], N[(N[(b * b), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision] + b), $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 (+.f64 (exp.f64 a) (exp.f64 b)) < 1.1000000000000001

   1. Initial program 9.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-exp.f64 5.5

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

   5. Simplified 5.5%

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

   6. Taylor expanded in b around inf

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

      1. distribute-lft-in N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. rgt-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 5.2

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

   8. Simplified 5.2%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       9. lower-*.f64 47.8

          \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot -0.16666666666666666}, b\right) \]

   11. Simplified 47.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)} \]

   if 1.1000000000000001 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 97.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 95.6

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

   5. Simplified 95.6%

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

   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 94.7

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

   8. Simplified 94.7%

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

Alternative 7: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(log1p(exp(a)) + Float64(b / Float64(1.0 + exp(a))))
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[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in b around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. lower-+.f64 N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 72.5

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

5. Simplified 72.5%

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

Alternative 8: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = 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 / (1.0 + Math.exp(a));
	} 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 / (1.0 + math.exp(a))
	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(1.0 + exp(a)));
	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[(1.0 + N[Exp[a], $MachinePrecision]), $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 8.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Simplified 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \mathsf{log1p}\left(e^{a}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   12. 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

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

   13. Simplified 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 63.5

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

   5. Simplified 63.5%

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

Alternative 9: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(Float64(b * Float64(fma(b, 0.5, 1.0) + Float64(exp(a) / b))))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[Log[1 + N[(b * N[(N[(b * 0.5 + 1.0), $MachinePrecision] + N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in b around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-exp.f64 49.6

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

5. Simplified 49.6%

   \[\leadsto \log \color{blue}{\left(1 + \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), e^{a}\right)\right)} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lower-log1p.f64 72.2

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), e^{a}\right)\right)} \]

7. Applied egg-rr 72.2%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), e^{a}\right)\right)} \]

8. Taylor expanded in b around inf

   \[\leadsto \mathsf{log1p}\left(\color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \left(\frac{1}{b} + \frac{e^{a}}{{b}^{2}}\right)\right)}\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \left(\frac{1}{b} + \frac{e^{a}}{{b}^{2}}\right)\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{b \cdot \left(b \cdot \left(\frac{1}{2} + \left(\frac{1}{b} + \frac{e^{a}}{{b}^{2}}\right)\right)\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{log1p}\left(b \cdot \color{blue}{\left(\left(\frac{1}{2} + \left(\frac{1}{b} + \frac{e^{a}}{{b}^{2}}\right)\right) \cdot b\right)}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{b \cdot \left(\left(\frac{1}{2} + \left(\frac{1}{b} + \frac{e^{a}}{{b}^{2}}\right)\right) \cdot b\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{log1p}\left(b \cdot \color{blue}{\left(b \cdot \left(\frac{1}{2} + \left(\frac{1}{b} + \frac{e^{a}}{{b}^{2}}\right)\right)\right)}\right) \]

   6. associate-+r+ N/A

      \[\leadsto \mathsf{log1p}\left(b \cdot \left(b \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{b}\right) + \frac{e^{a}}{{b}^{2}}\right)}\right)\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{log1p}\left(b \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{b}\right) \cdot b + \frac{e^{a}}{{b}^{2}} \cdot b\right)}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{log1p}\left(b \cdot \left(\color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{b}\right)} + \frac{e^{a}}{{b}^{2}} \cdot b\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{log1p}\left(b \cdot \left(b \cdot \color{blue}{\left(\frac{1}{b} + \frac{1}{2}\right)} + \frac{e^{a}}{{b}^{2}} \cdot b\right)\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto \mathsf{log1p}\left(b \cdot \left(\color{blue}{\left(\frac{1}{b} \cdot b + \frac{1}{2} \cdot b\right)} + \frac{e^{a}}{{b}^{2}} \cdot b\right)\right) \]

   11. lft-mult-inverse N/A

       \[\leadsto \mathsf{log1p}\left(b \cdot \left(\left(\color{blue}{1} + \frac{1}{2} \cdot b\right) + \frac{e^{a}}{{b}^{2}} \cdot b\right)\right) \]

   12. associate-/r/ N/A

       \[\leadsto \mathsf{log1p}\left(b \cdot \left(\left(1 + \frac{1}{2} \cdot b\right) + \color{blue}{\frac{e^{a}}{\frac{{b}^{2}}{b}}}\right)\right) \]

   13. *-rgt-identity N/A

       \[\leadsto \mathsf{log1p}\left(b \cdot \left(\left(1 + \frac{1}{2} \cdot b\right) + \frac{e^{a}}{\frac{\color{blue}{{b}^{2} \cdot 1}}{b}}\right)\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{log1p}\left(b \cdot \left(\left(1 + \frac{1}{2} \cdot b\right) + \frac{e^{a}}{\color{blue}{{b}^{2} \cdot \frac{1}{b}}}\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{log1p}\left(b \cdot \left(\left(1 + \frac{1}{2} \cdot b\right) + \frac{e^{a}}{\color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{b}}\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{log1p}\left(b \cdot \left(\left(1 + \frac{1}{2} \cdot b\right) + \frac{e^{a}}{\color{blue}{b \cdot \left(b \cdot \frac{1}{b}\right)}}\right)\right) \]

   17. rgt-mult-inverse N/A

       \[\leadsto \mathsf{log1p}\left(b \cdot \left(\left(1 + \frac{1}{2} \cdot b\right) + \frac{e^{a}}{b \cdot \color{blue}{1}}\right)\right) \]

   18. *-rgt-identity N/A

       \[\leadsto \mathsf{log1p}\left(b \cdot \left(\left(1 + \frac{1}{2} \cdot b\right) + \frac{e^{a}}{\color{blue}{b}}\right)\right) \]

10. Simplified 72.2%

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

Alternative 10: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, 0.5, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-9)
   (fma (* b b) (* b -0.16666666666666666) b)
   (fma b 0.5 (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-9) {
		tmp = fma((b * b), (b * -0.16666666666666666), b);
	} else {
		tmp = fma(b, 0.5, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-9)
		tmp = fma(Float64(b * b), Float64(b * -0.16666666666666666), b);
	else
		tmp = fma(b, 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], 2e-9], N[(N[(b * b), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision] + b), $MachinePrecision], N[(b * 0.5 + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.00000000000000012e-9

   1. Initial program 9.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-exp.f64 8.1

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

   5. Simplified 8.1%

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

   6. Taylor expanded in b around inf

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

      1. distribute-lft-in N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. rgt-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 7.5

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

   8. Simplified 7.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       9. lower-*.f64 95.7

          \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot -0.16666666666666666}, b\right) \]

   11. Simplified 95.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)} \]

   if 2.00000000000000012e-9 < (exp.f64 a)

   1. Initial program 66.8%

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

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

   5. Simplified 63.7%

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

   6. Taylor expanded in b 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 61.7

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

   8. Simplified 61.7%

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

Alternative 11: 96.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), e^{a}\right)\right) \end{array} \]

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return log1p(fma(b, fma(b, 0.5, 1.0), exp(a)));
}

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(fma(b, fma(b, 0.5, 1.0), exp(a)))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[Log[1 + N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in b around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-exp.f64 49.6

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

5. Simplified 49.6%

   \[\leadsto \log \color{blue}{\left(1 + \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), e^{a}\right)\right)} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lower-log1p.f64 72.2

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), e^{a}\right)\right)} \]

7. Applied egg-rr 72.2%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), e^{a}\right)\right)} \]
8. Add Preprocessing

Alternative 12: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-9)
   (fma (* b b) (* b -0.16666666666666666) b)
   (log1p (+ b 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-9) {
		tmp = fma((b * b), (b * -0.16666666666666666), b);
	} else {
		tmp = log1p((b + 1.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-9)
		tmp = fma(Float64(b * b), Float64(b * -0.16666666666666666), b);
	else
		tmp = log1p(Float64(b + 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], 2e-9], N[(N[(b * b), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision] + b), $MachinePrecision], N[Log[1 + N[(b + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.00000000000000012e-9

   1. Initial program 9.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-exp.f64 8.1

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

   5. Simplified 8.1%

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

   6. Taylor expanded in b around inf

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

      1. distribute-lft-in N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. rgt-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 7.5

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

   8. Simplified 7.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       9. lower-*.f64 95.7

          \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot -0.16666666666666666}, b\right) \]

   11. Simplified 95.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)} \]

   if 2.00000000000000012e-9 < (exp.f64 a)

   1. Initial program 66.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 62.3

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

   5. Simplified 62.3%

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

      1. lift-exp.f64 N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-+.f64 62.3

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

   7. Applied egg-rr 62.3%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 60.7

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

   10. Simplified 60.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-9)
   (fma (* b b) (* b -0.16666666666666666) b)
   (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-9) {
		tmp = fma((b * b), (b * -0.16666666666666666), b);
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-9)
		tmp = fma(Float64(b * b), Float64(b * -0.16666666666666666), b);
	else
		tmp = log(Float64(b + 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], 2e-9], N[(N[(b * b), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision] + b), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.00000000000000012e-9

   1. Initial program 9.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-exp.f64 8.1

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

   5. Simplified 8.1%

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

   6. Taylor expanded in b around inf

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

      1. distribute-lft-in N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. rgt-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 7.5

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

   8. Simplified 7.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       9. lower-*.f64 95.7

          \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot -0.16666666666666666}, b\right) \]

   11. Simplified 95.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)} \]

   if 2.00000000000000012e-9 < (exp.f64 a)

   1. Initial program 66.8%

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

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

   5. Simplified 63.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 60.7

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

   8. Simplified 60.7%

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

Alternative 14: 94.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-9) (fma (* b b) (* b -0.16666666666666666) b) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-9) {
		tmp = fma((b * b), (b * -0.16666666666666666), b);
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-9)
		tmp = fma(Float64(b * b), Float64(b * -0.16666666666666666), b);
	else
		tmp = log1p(1.0);
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-9], N[(N[(b * b), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision] + b), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.00000000000000012e-9

   1. Initial program 9.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-exp.f64 8.1

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

   5. Simplified 8.1%

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

   6. Taylor expanded in b around inf

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

      1. distribute-lft-in N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. rgt-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 7.5

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

   8. Simplified 7.5%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

       9. lower-*.f64 95.7

          \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot -0.16666666666666666}, b\right) \]

   11. Simplified 95.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)} \]

   if 2.00000000000000012e-9 < (exp.f64 a)

   1. Initial program 66.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 63.5

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

   5. Simplified 63.5%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 61.9%

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

Alternative 15: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(Float64(b + exp(a)))
end

Wolfram

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

Derivation

1. Initial program 52.4%

   \[\log \left(e^{a} + e^{b}\right) \]
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 48.6

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

5. Simplified 48.6%

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

   1. lift-exp.f64 N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lower-log1p.f64 N/A

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

   6. lower-+.f64 71.0

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

7. Applied egg-rr 71.0%

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

8. Final simplification 71.0%

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

Alternative 16: 50.2% accurate, 13.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{fma}\left(\mathsf{fma}\left(b, 0.125, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (fma (fma b 0.125 -0.16666666666666666) (* b (* b b)) b))

C

assert(a < b);
double code(double a, double b) {
	return fma(fma(b, 0.125, -0.16666666666666666), (b * (b * b)), b);
}

Julia

a, b = sort([a, b])
function code(a, b)
	return fma(fma(b, 0.125, -0.16666666666666666), Float64(b * Float64(b * b)), b)
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(b * 0.125 + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in b around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-exp.f64 49.6

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

5. Simplified 49.6%

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

6. Taylor expanded in b around inf

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

   1. distribute-lft-in N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. rgt-mult-inverse N/A

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

   5. *-rgt-identity N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 4.3

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

8. Simplified 4.3%

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

9. Taylor expanded in b around 0

   \[\leadsto \color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left(\frac{1}{8} \cdot b - \frac{1}{6}\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto b \cdot \color{blue}{\left({b}^{2} \cdot \left(\frac{1}{8} \cdot b - \frac{1}{6}\right) + 1\right)} \]

    2. distribute-rgt-in N/A

       \[\leadsto \color{blue}{\left({b}^{2} \cdot \left(\frac{1}{8} \cdot b - \frac{1}{6}\right)\right) \cdot b + 1 \cdot b} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left(\frac{1}{8} \cdot b - \frac{1}{6}\right) \cdot {b}^{2}\right)} \cdot b + 1 \cdot b \]

    4. associate-*l* N/A

       \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot b - \frac{1}{6}\right) \cdot \left({b}^{2} \cdot b\right)} + 1 \cdot b \]

    5. unpow2 N/A

       \[\leadsto \left(\frac{1}{8} \cdot b - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) + 1 \cdot b \]

    6. unpow3 N/A

       \[\leadsto \left(\frac{1}{8} \cdot b - \frac{1}{6}\right) \cdot \color{blue}{{b}^{3}} + 1 \cdot b \]

    7. *-lft-identity N/A

       \[\leadsto \left(\frac{1}{8} \cdot b - \frac{1}{6}\right) \cdot {b}^{3} + \color{blue}{b} \]

    8. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot b - \frac{1}{6}, {b}^{3}, b\right)} \]

    9. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{8} \cdot b + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {b}^{3}, b\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot \frac{1}{8}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {b}^{3}, b\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(b \cdot \frac{1}{8} + \color{blue}{\frac{-1}{6}}, {b}^{3}, b\right) \]

    12. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(b, \frac{1}{8}, \frac{-1}{6}\right)}, {b}^{3}, b\right) \]

    13. cube-mult N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, \frac{1}{8}, \frac{-1}{6}\right), \color{blue}{b \cdot \left(b \cdot b\right)}, b\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, \frac{1}{8}, \frac{-1}{6}\right), b \cdot \color{blue}{{b}^{2}}, b\right) \]

    15. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, \frac{1}{8}, \frac{-1}{6}\right), \color{blue}{b \cdot {b}^{2}}, b\right) \]

    16. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, \frac{1}{8}, \frac{-1}{6}\right), b \cdot \color{blue}{\left(b \cdot b\right)}, b\right) \]

    17. lower-*.f64 26.4

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b, 0.125, -0.16666666666666666\right), b \cdot \color{blue}{\left(b \cdot b\right)}, b\right) \]

11. Simplified 26.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.125, -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)} \]
12. Add Preprocessing

Alternative 17: 50.0% accurate, 17.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (fma (* b b) (* b -0.16666666666666666) b))

C

assert(a < b);
double code(double a, double b) {
	return fma((b * b), (b * -0.16666666666666666), b);
}

Julia

a, b = sort([a, b])
function code(a, b)
	return fma(Float64(b * b), Float64(b * -0.16666666666666666), b)
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(b * b), $MachinePrecision] * N[(b * -0.16666666666666666), $MachinePrecision] + b), $MachinePrecision]

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in b around 0

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

   1. lower-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-exp.f64 49.6

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

5. Simplified 49.6%

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

6. Taylor expanded in b around inf

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

   1. distribute-lft-in N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. rgt-mult-inverse N/A

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

   5. *-rgt-identity N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 4.3

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

8. Simplified 4.3%

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

9. Taylor expanded in b around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. *-rgt-identity N/A

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

    6. lower-fma.f64 N/A

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

    7. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{b \cdot b}, b \cdot \frac{-1}{6}, b\right) \]

    9. lower-*.f64 26.4

       \[\leadsto \mathsf{fma}\left(b \cdot b, \color{blue}{b \cdot -0.16666666666666666}, b\right) \]

11. Simplified 26.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, b \cdot -0.16666666666666666, b\right)} \]
12. Add Preprocessing

Alternative 18: 12.0% accurate, 50.7× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return b * 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 = b * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

   \[\log \left(e^{a} + e^{b}\right) \]
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 48.7

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

5. Simplified 48.7%

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

6. Taylor expanded in b 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-log.f64 47.2

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

8. Simplified 47.2%

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

9. Taylor expanded in b around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 7.3

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

11. Simplified 7.3%

    \[\leadsto \color{blue}{b \cdot 0.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))