symmetry log of sum of exp

Percentage Accurate: 53.7% → 98.9%

Time: 12.6s

Alternatives: 19

Speedup: 2.8×

• could not determine a ground truth

  1. a = 1.1768115810034397e+45
  2. b = 2.0625270574729547e+20

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: 53.7% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -37:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -37.0) (/ b (+ 1.0 (exp a))) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -37.0) {
		tmp = b / (1.0 + exp(a));
	} 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 (a <= (-37.0d0)) then
        tmp = b / (1.0d0 + exp(a))
    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 (a <= -37.0) {
		tmp = b / (1.0 + Math.exp(a));
	} 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 a <= -37.0:
		tmp = b / (1.0 + math.exp(a))
	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 (a <= -37.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	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 (a <= -37.0)
		tmp = b / (1.0 + exp(a));
	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[a, -37.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -37

   1. Initial program 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 100.0%

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

   if -37 < a

   1. Initial program 70.5%

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

Alternative 2: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 74.7%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.125, 0.5\right), \mathsf{log1p}\left(e^{a}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ 1.0 (exp a)))
   (fma b (fma b 0.125 0.5) (log1p (exp a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = fma(b, fma(b, 0.125, 0.5), log1p(exp(a)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = fma(b, fma(b, 0.125, 0.5), log1p(exp(a)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(b, \frac{1}{2} + \color{blue}{\left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 66.2%

      \[\leadsto \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{0.125}, 0.5\right), \mathsf{log1p}\left(e^{a}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

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

   2. associate-*r/ N/A

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

   3. lower-+.f64 N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 74.3

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

5. Applied rewrites 74.3%

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

Alternative 5: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} + e^{b} \leq 1.2:\\ \;\;\;\;b \cdot \mathsf{fma}\left(b, 0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, 0.5, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (+ (exp a) (exp b)) 1.2)
   (* b (fma b 0.125 0.5))
   (fma b 0.5 (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if ((exp(a) + exp(b)) <= 1.2) {
		tmp = b * fma(b, 0.125, 0.5);
	} else {
		tmp = fma(b, 0.5, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) + exp(b)) <= 1.2)
		tmp = Float64(b * fma(b, 0.125, 0.5));
	else
		tmp = fma(b, 0.5, log(2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision], 1.2], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision]), $MachinePrecision], N[(b * 0.5 + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 a) (exp.f64 b)) < 1.19999999999999996

   1. Initial program 9.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 10.7%

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

   if 1.19999999999999996 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 95.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 92.5

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

   7. Applied rewrites 92.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 90.5%

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

Alternative 6: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 67.4

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

   5. Applied rewrites 67.4%

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

Alternative 7: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.125, 0.5\right), \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot -0.03125, 0.125\right), \mathsf{fma}\left(b, -0.25, 0.5\right)\right), \log 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ 1.0 (exp a)))
   (fma
    b
    (fma b 0.125 0.5)
    (fma a (fma a (fma b (* b -0.03125) 0.125) (fma b -0.25 0.5)) (log 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = fma(b, fma(b, 0.125, 0.5), fma(a, fma(a, fma(b, (b * -0.03125), 0.125), fma(b, -0.25, 0.5)), log(2.0)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = fma(b, fma(b, 0.125, 0.5), fma(a, fma(a, fma(b, Float64(b * -0.03125), 0.125), fma(b, -0.25, 0.5)), log(2.0)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision] + N[(a * N[(a * N[(b * N[(b * -0.03125), $MachinePrecision] + 0.125), $MachinePrecision] + N[(b * -0.25 + 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in a around 0

      \[\leadsto \log 2 + \color{blue}{\left(a \cdot \left(\frac{1}{2} + \left(a \cdot \left(\frac{1}{8} + b \cdot \left(\frac{-1}{2} \cdot \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{-1}{4} \cdot b + \frac{3}{16} \cdot b\right)\right)\right) + b \cdot \left(\left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right) - \frac{1}{4}\right)\right)\right) + b \cdot \left(\frac{1}{2} + \left(\frac{-1}{8} \cdot b + \frac{1}{4} \cdot b\right)\right)\right)} \]

   8. Applied rewrites 66.0%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.125, 0.5\right)}, \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(b, b \cdot -0.03125, 0.125\right), \mathsf{fma}\left(b, -0.25, 0.5\right)\right), \log 2\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0) + fma(b, fma(b, 0.5, 1.0), 1.0)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0) + fma(b, fma(b, 0.5, 1.0), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 65.8

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

   8. Applied rewrites 65.8%

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

Alternative 9: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((fma(b, fma(b, 0.5, 1.0), 1.0) + fma(a, fma(a, 0.5, 1.0), 1.0)));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(a * N[(a * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 65.7

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

   8. Applied rewrites 65.7%

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

4. Final simplification 74.1%

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

Alternative 10: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -37:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -37.0)
   (/ b (+ 1.0 (exp a)))
   (log (+ (exp a) (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -37.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((exp(a) + fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -37.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(a) + fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -37.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -37

   1. Initial program 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 100.0%

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

   if -37 < a

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 65.8

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

   5. Applied rewrites 65.8%

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

Alternative 11: 97.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((fma(b, fma(b, 0.5, 1.0), 1.0) + (a + 1.0)));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 65.5

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

   8. Applied rewrites 65.5%

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

4. Final simplification 74.0%

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

Alternative 12: 97.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = fma(b, fma(b, 0.125, 0.5), log(2.0));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 9.0

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.3%

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

Alternative 13: 56.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -105:\\ \;\;\;\;b \cdot \mathsf{fma}\left(b, 0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.125, 0.5\right), \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -105.0)
   (* b (fma b 0.125 0.5))
   (fma b (fma b 0.125 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -105.0) {
		tmp = b * fma(b, 0.125, 0.5);
	} else {
		tmp = fma(b, fma(b, 0.125, 0.5), log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -105.0)
		tmp = Float64(b * fma(b, 0.125, 0.5));
	else
		tmp = fma(b, fma(b, 0.125, 0.5), log(2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -105.0], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -105

   1. Initial program 8.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 4.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 18.7%

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

   if -105 < a

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.3%

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

Alternative 14: 56.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -105:\\ \;\;\;\;b \cdot \mathsf{fma}\left(b, 0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -105.0) (* b (fma b 0.125 0.5)) (log1p (+ b 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -105.0) {
		tmp = b * fma(b, 0.125, 0.5);
	} else {
		tmp = log1p((b + 1.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -105.0)
		tmp = Float64(b * fma(b, 0.125, 0.5));
	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, -105.0], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(b + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -105

   1. Initial program 8.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 4.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 18.7%

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

   if -105 < a

   1. Initial program 70.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 67.4

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 63.9%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -105:\\ \;\;\;\;b \cdot \mathsf{fma}\left(b, 0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;b \cdot \mathsf{fma}\left(b, 0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.0) (* b (fma b 0.125 0.5)) (log1p (+ a 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b * fma(b, 0.125, 0.5);
	} else {
		tmp = log1p((a + 1.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.0)
		tmp = Float64(b * fma(b, 0.125, 0.5));
	else
		tmp = log1p(Float64(a + 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, -1.0], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(a + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 10.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 4.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 18.4%

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

   if -1 < a

   1. Initial program 70.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 65.3

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 64.8%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;b \cdot \mathsf{fma}\left(b, 0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -105:\\ \;\;\;\;b \cdot \mathsf{fma}\left(b, 0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -105.0) (* b (fma b 0.125 0.5)) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -105.0) {
		tmp = b * fma(b, 0.125, 0.5);
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -105.0)
		tmp = Float64(b * fma(b, 0.125, 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, -105.0], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision]), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -105

   1. Initial program 8.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 4.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 18.7%

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

   if -105 < a

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 64.1%

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

Alternative 17: 11.8% accurate, 25.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ b \cdot \mathsf{fma}\left(b, 0.125, 0.5\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* b (fma b 0.125 0.5)))

C

assert(a < b);
double code(double a, double b) {
	return b * fma(b, 0.125, 0.5);
}

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b * fma(b, 0.125, 0.5))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(b * N[(b * 0.125 + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 50.3%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 7.5%

    \[\leadsto b \cdot \mathsf{fma}\left(b, \color{blue}{0.125}, 0.5\right) \]
12. Add Preprocessing

Alternative 18: 5.2% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * (b * 0.125d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 74.7%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 50.3%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 4.5%

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

Alternative 19: 3.2% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a * (a * 0.125d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 50.2

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

5. Applied rewrites 50.2%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 49.3%

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

9. Taylor expanded in a around inf

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

11. Applied rewrites 4.2%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))