symmetry log of sum of exp

Percentage Accurate: 53.0% → 98.2%

Time: 12.7s

Alternatives: 17

Speedup: 2.9×

• could not determine a ground truth

  1. a = 2.5069690633928617e+240
  2. b = 7.7774655165067485e-292

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: 53.0% 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: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.2e+94)
   (/ b (+ (exp a) 1.0))
   (if (<= a -3.1e-17) (log1p (+ (exp a) b)) (log1p (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.2e+94) {
		tmp = b / (exp(a) + 1.0);
	} else if (a <= -3.1e-17) {
		tmp = log1p((exp(a) + b));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.2e+94) {
		tmp = b / (Math.exp(a) + 1.0);
	} else if (a <= -3.1e-17) {
		tmp = Math.log1p((Math.exp(a) + b));
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.2e+94:
		tmp = b / (math.exp(a) + 1.0)
	elif a <= -3.1e-17:
		tmp = math.log1p((math.exp(a) + b))
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.2e+94)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	elseif (a <= -3.1e-17)
		tmp = log1p(Float64(exp(a) + b));
	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[a, -2.2e+94], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -3.1e-17], N[Log[1 + N[(N[Exp[a], $MachinePrecision] + b), $MachinePrecision]], $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.20000000000000012e94

   1. Initial program 10.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

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

   if -2.20000000000000012e94 < a < -3.0999999999999998e-17

   1. Initial program 21.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{\left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot b\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 21.0%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 21.0%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. log1p-define N/A

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

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a} + b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 95.8%

          \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 95.8%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{b}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 95.8%

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

   if -3.0999999999999998e-17 < a

   1. Initial program 67.2%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{b}\right)\right) \]

      3. exp-lowering-exp.f64 64.9%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(b\right)\right) \]

   5. Simplified 64.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.exp(a) + 1.0
	return math.log1p(math.exp(a)) + (b * (((1.0 + (b * 0.5)) * (1.0 / t_0)) + (b * (-0.5 / math.pow(t_0, 2.0)))))

Julia

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in b around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)\right)}\right) \]

   2. log1p-define N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)\right)\right) \]

   3. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)\right)\right) \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)}\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \left(\frac{1}{1 + e^{a}} + \color{blue}{\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right)}\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \left(\frac{1}{1 + e^{a}} + \left(\frac{1}{2} \cdot b\right) \cdot \color{blue}{\left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)}\right)\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \left(\frac{1}{1 + e^{a}} + \left(\frac{1}{2} \cdot b\right) \cdot \left(\frac{1}{1 + e^{a}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right)}\right)\right)\right)\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \left(\frac{1}{1 + e^{a}} + \left(\frac{1}{1 + e^{a}} \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) \cdot \left(\frac{1}{2} \cdot b\right)}\right)\right)\right)\right) \]

5. Simplified 76.1%

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

6. Final simplification 76.1%

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

Alternative 3: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1

   1. Initial program 49.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 76.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 76.1%

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

   if 1 < (exp.f64 a)

   1. Initial program 75.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{b}\right)\right) \]

      3. exp-lowering-exp.f64 31.0%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(b\right)\right) \]

   5. Simplified 31.0%

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

4. Final simplification 74.5%

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

Alternative 4: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-21) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-21) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((exp(a) + exp(b)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 5d-21) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + exp(b)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-21) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((Math.exp(a) + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-21:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((math.exp(a) + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-21)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(exp(a) + exp(b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-21)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((exp(a) + exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-21], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999973e-21

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 97.6%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 97.6%

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

   if 4.99999999999999973e-21 < (exp.f64 a)

   1. Initial program 67.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-21) (/ b (+ (exp a) 1.0)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-21) {
		tmp = b / (exp(a) + 1.0);
	} 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) <= 5e-21) {
		tmp = b / (Math.exp(a) + 1.0);
	} 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) <= 5e-21:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-21)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-21], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999973e-21

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 97.6%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 97.6%

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

   if 4.99999999999999973e-21 < (exp.f64 a)

   1. Initial program 67.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{b}\right)\right) \]

      3. exp-lowering-exp.f64 64.1%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(b\right)\right) \]

   5. Simplified 64.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b \cdot \left(1 + b \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3e+94)
   (/ b (+ (exp a) 1.0))
   (log1p (+ (exp a) (* b (+ 1.0 (* b 0.5)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3e+94) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p((exp(a) + (b * (1.0 + (b * 0.5)))));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3e+94) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p((Math.exp(a) + (b * (1.0 + (b * 0.5)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3e+94:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p((math.exp(a) + (b * (1.0 + (b * 0.5)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3e+94)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(Float64(exp(a) + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -3e+94], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(N[Exp[a], $MachinePrecision] + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.0000000000000001e94

   1. Initial program 10.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

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

   if -3.0000000000000001e94 < a

   1. Initial program 61.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{\left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot b\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 60.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 60.1%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. log1p-define N/A

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

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a} + b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 69.2%

          \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a} + b \cdot \left(1 + b \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b \cdot \left(1 + b \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-21)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* a (+ 0.5 (* a 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-21) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (a * (0.5 + (a * 0.125)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 5d-21) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (a * (0.5d0 + (a * 0.125d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-21) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * (0.5 + (a * 0.125)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-21:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (a * (0.5 + (a * 0.125)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-21)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(a * Float64(0.5 + Float64(a * 0.125))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-21)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (a * (0.5 + (a * 0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-21], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * N[(0.5 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999973e-21

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 97.6%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 97.6%

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

   if 4.99999999999999973e-21 < (exp.f64 a)

   1. Initial program 67.9%

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

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.5%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.5%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(a \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot a\right)\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{a} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot a\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{2} + \frac{1}{8} \cdot a\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{8} \cdot a\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{8}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 63.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{8}}\right)\right)\right)\right) \]

   8. Simplified 63.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-21) (/ b (+ (exp a) 1.0)) (+ (* b 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-21) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = (b * 0.5) + log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 5d-21) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = (b * 0.5d0) + log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-21) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-21:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-21)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(b * 0.5) + log(2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-21)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = (b * 0.5) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-21], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999973e-21

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 97.6%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 97.6%

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

   if 4.99999999999999973e-21 < (exp.f64 a)

   1. Initial program 67.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(1 + e^{b}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right) \]

      2. exp-lowering-exp.f64 64.0%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 64.0%

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

   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

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot b\right), \color{blue}{\log 2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \frac{1}{2}\right), \log \color{blue}{2}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \log \color{blue}{2}\right) \]

      5. log-lowering-log.f64 62.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{log.f64}\left(2\right)\right) \]

   8. Simplified 62.2%

      \[\leadsto \color{blue}{b \cdot 0.5 + \log 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -90:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -90.0) (/ b (+ (exp a) 1.0)) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -90.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -90.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -90.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -90.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(a));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -90.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -90

   1. Initial program 9.4%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 98.8%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 98.8%

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

   if -90 < a

   1. Initial program 67.5%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.5%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -90:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -42:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -42.0)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* b (+ 0.5 (* b 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -42.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (b * (0.5 + (b * 0.125)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-42.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * (0.5d0 + (b * 0.125d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -42.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * (0.5 + (b * 0.125)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -42.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (b * (0.5 + (b * 0.125)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -42.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(b * Float64(0.5 + Float64(b * 0.125))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -42.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (b * (0.5 + (b * 0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -42.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -42

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 97.6%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 97.6%

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

   if -42 < a

   1. Initial program 67.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)\right)}\right) \]

      2. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)\right)\right) \]

      3. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)\right)\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \left(\frac{1}{1 + e^{a}} + \color{blue}{\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right)}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \left(\frac{1}{1 + e^{a}} + \left(\frac{1}{2} \cdot b\right) \cdot \color{blue}{\left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \left(\frac{1}{1 + e^{a}} + \left(\frac{1}{2} \cdot b\right) \cdot \left(\frac{1}{1 + e^{a}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right)}\right)\right)\right)\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \left(\frac{1}{1 + e^{a}} + \left(\frac{1}{1 + e^{a}} \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) \cdot \left(\frac{1}{2} \cdot b\right)}\right)\right)\right)\right) \]

   5. Simplified 66.1%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{b} \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right)\right) \]

      3. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(b \cdot \left(\frac{1}{2} + b \cdot \color{blue}{\left(\frac{-1}{8} + \frac{1}{4}\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{8}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \color{blue}{b}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{8} \cdot b\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{8} \cdot b\right)}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{8}}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 62.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{8}}\right)\right)\right)\right) \]

   8. Simplified 62.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -42:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 94.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -34:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -34.0) (log1p b) (+ (* b 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -34.0) {
		tmp = log1p(b);
	} else {
		tmp = (b * 0.5) + log(2.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -34.0) {
		tmp = Math.log1p(b);
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -34.0:
		tmp = math.log1p(b)
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -34.0)
		tmp = log1p(b);
	else
		tmp = Float64(Float64(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[a, -34.0], N[Log[1 + b], $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -34

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{\left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot b\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 8.4%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 8.4%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. log1p-define N/A

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

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a} + b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 95.1%

          \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{b}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 93.6%

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

   11. Taylor expanded in b around inf

       \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{b}\right) \]
   12. Step-by-step derivation

   13. Simplified 92.5%

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

   if -34 < a

   1. Initial program 67.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(1 + e^{b}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right) \]

      2. exp-lowering-exp.f64 64.0%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 64.0%

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

   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

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot b\right), \color{blue}{\log 2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \frac{1}{2}\right), \log \color{blue}{2}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \log \color{blue}{2}\right) \]

      5. log-lowering-log.f64 62.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{log.f64}\left(2\right)\right) \]

   8. Simplified 62.2%

      \[\leadsto \color{blue}{b \cdot 0.5 + \log 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 94.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -42:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -42.0) (log1p b) (log1p (+ b 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -42.0) {
		tmp = log1p(b);
	} else {
		tmp = log1p((b + 1.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -42.0) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log1p((b + 1.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -42.0:
		tmp = math.log1p(b)
	else:
		tmp = math.log1p((b + 1.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -42.0)
		tmp = log1p(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[a, -42.0], N[Log[1 + b], $MachinePrecision], N[Log[1 + N[(b + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -42

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{\left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot b\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 8.4%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 8.4%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. log1p-define N/A

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

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a} + b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 95.1%

          \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{b}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 93.6%

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

   11. Taylor expanded in b around inf

       \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{b}\right) \]
   12. Step-by-step derivation

   13. Simplified 92.5%

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

   if -42 < a

   1. Initial program 67.9%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{\left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot b\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 66.2%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 66.2%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. log1p-define N/A

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

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a} + b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 66.3%

          \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 66.3%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{b}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 64.8%

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

   11. Taylor expanded in a around 0

       \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{\left(1 + b\right)}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{log1p.f64}\left(\left(b + 1\right)\right) \]

       2. +-lowering-+.f64 61.3%

          \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(b, 1\right)\right) \]

   13. Simplified 61.3%

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

Alternative 13: 94.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -46:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -46.0) (log1p b) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -46.0) {
		tmp = log1p(b);
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -46.0) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -46.0:
		tmp = math.log1p(b)
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -46.0)
		tmp = log1p(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[a, -46.0], N[Log[1 + b], $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -46

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{\left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot b\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 8.4%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 8.4%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. log1p-define N/A

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

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a} + b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 95.1%

          \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{b}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 93.6%

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

   11. Taylor expanded in b around inf

       \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{b}\right) \]
   12. Step-by-step derivation

   13. Simplified 92.5%

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

   if -46 < a

   1. Initial program 67.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(1 + e^{b}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right) \]

      2. exp-lowering-exp.f64 64.0%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 64.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(2 + b\right)}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(b + 2\right)\right) \]

      2. +-lowering-+.f64 61.3%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(b, 2\right)\right) \]

   8. Simplified 61.3%

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

Alternative 14: 94.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -43:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -43.0) (log1p b) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -43.0) {
		tmp = log1p(b);
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -43.0) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log1p(1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -43.0:
		tmp = math.log1p(b)
	else:
		tmp = math.log1p(1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -43.0)
		tmp = log1p(b);
	else
		tmp = log1p(1.0);
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -43.0], N[Log[1 + b], $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -43

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{\left(1 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot b\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 8.4%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 8.4%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. log1p-define N/A

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

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a} + b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(b \cdot \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \left(1 + b \cdot \frac{1}{2}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 95.1%

          \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 95.1%

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

   8. Taylor expanded in b around 0

      \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{b}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 93.6%

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

   11. Taylor expanded in b around inf

       \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{b}\right) \]
   12. Step-by-step derivation

   13. Simplified 92.5%

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

   if -43 < a

   1. Initial program 67.9%

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

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.5%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 62.1%

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

Alternative 15: 55.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -46:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -46.0) (* b 0.5) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -46.0) {
		tmp = b * 0.5;
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -46.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log1p(1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -46.0:
		tmp = b * 0.5
	else:
		tmp = math.log1p(1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -46.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log1p(1.0);
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -46.0], N[(b * 0.5), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -46

   1. Initial program 9.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(1 + e^{b}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right) \]

      2. exp-lowering-exp.f64 4.3%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 4.3%

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

   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

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot b\right), \color{blue}{\log 2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \frac{1}{2}\right), \log \color{blue}{2}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \log \color{blue}{2}\right) \]

      5. log-lowering-log.f64 4.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{log.f64}\left(2\right)\right) \]

   8. Simplified 4.8%

      \[\leadsto \color{blue}{b \cdot 0.5 + \log 2} \]

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 18.5%

          \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{2}}\right) \]

   11. Simplified 18.5%

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

   if -46 < a

   1. Initial program 67.9%

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

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.5%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.5%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 62.1%

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

Alternative 16: 12.1% accurate, 101.0× 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 50.0%

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

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(1 + e^{b}\right)}\right) \]
4. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right) \]

   2. exp-lowering-exp.f64 45.8%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

5. Simplified 45.8%

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

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

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot b\right), \color{blue}{\log 2}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(b \cdot \frac{1}{2}\right), \log \color{blue}{2}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \log \color{blue}{2}\right) \]

   5. log-lowering-log.f64 44.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \frac{1}{2}\right), \mathsf{log.f64}\left(2\right)\right) \]

8. Simplified 44.7%

   \[\leadsto \color{blue}{b \cdot 0.5 + \log 2} \]

9. Taylor expanded in b around inf

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

    1. *-commutative N/A

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

    2. *-lowering-*.f64 8.0%

       \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{2}}\right) \]

11. Simplified 8.0%

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

Alternative 17: 2.6% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in b around 0

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

   1. log1p-define N/A

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

   2. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

   3. exp-lowering-exp.f64 47.2%

      \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

5. Simplified 47.2%

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

6. Taylor expanded in a around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{\frac{1}{2}} \cdot a\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(a \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

   4. *-lowering-*.f64 44.5%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{2}}\right)\right) \]

8. Simplified 44.5%

   \[\leadsto \color{blue}{\log 2 + a \cdot 0.5} \]

9. Taylor expanded in a around inf

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

    1. *-lowering-*.f64 7.2%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{a}\right) \]

11. Simplified 7.2%

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

12. Final simplification 7.2%

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

Reproduce

herbie shell --seed 2024164 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))