symmetry log of sum of exp

Percentage Accurate: 54.2% → 98.5%

Time: 10.9s

Alternatives: 12

Speedup: 1.5×

• could not determine a ground truth

  1. a = 2.272777065303425e+41
  2. b = -9.531476192868882e-284

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% 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: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(exp(a) + 1.0)
	return fma(fma(fma(0.5, b, 1.0), (t_0 ^ -1.0), Float64(Float64(-0.5 / (t_0 ^ 2.0)) * b)), b, 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[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[(N[(0.5 * b + 1.0), $MachinePrecision] * N[Power[t$95$0, -1.0], $MachinePrecision] + N[(N[(-0.5 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * b + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\log \left(1 + e^{a}\right) + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)} \]
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. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

5. Applied rewrites 78.0%

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

6. Final simplification 78.0%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. associate-*r/ N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-exp.f64 77.7

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

5. Applied rewrites 77.7%

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

Alternative 3: 58.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \mathsf{log1p}\left(e^{a}\right)\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (fma (fma 0.125 b 0.5) b (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	return fma(fma(0.125, b, 0.5), b, log1p(exp(a)));
}

Julia

a, b = sort([a, b])
function code(a, b)
	return fma(fma(0.125, b, 0.5), b, log1p(exp(a)))
end

Wolfram

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\log \left(1 + e^{a}\right) + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)} \]
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. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

5. Applied rewrites 78.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 57.5%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \mathsf{log1p}\left(e^{a}\right)\right) \]
9. Add Preprocessing

Alternative 4: 52.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.2e-156) (log1p (exp a)) (log (+ 1.0 (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.2e-156) {
		tmp = log1p(exp(a));
	} else {
		tmp = log((1.0 + exp(b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2.2e-156) {
		tmp = Math.log1p(Math.exp(a));
	} else {
		tmp = Math.log((1.0 + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 2.2e-156:
		tmp = math.log1p(math.exp(a))
	else:
		tmp = math.log((1.0 + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2.2e-156)
		tmp = log1p(exp(a));
	else
		tmp = log(Float64(1.0 + 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[b, 2.2e-156], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[Log[N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1999999999999999e-156

   1. Initial program 54.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 51.9

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

   5. Applied rewrites 51.9%

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

   if 2.1999999999999999e-156 < b

   1. Initial program 68.2%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 67.8%

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

Alternative 5: 52.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.2e-156) (log1p (exp a)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.2e-156) {
		tmp = log1p(exp(a));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 2.2e-156) {
		tmp = Math.log1p(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 b <= 2.2e-156:
		tmp = math.log1p(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 (b <= 2.2e-156)
		tmp = log1p(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[b, 2.2e-156], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1999999999999999e-156

   1. Initial program 54.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 51.9

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

   5. Applied rewrites 51.9%

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

   if 2.1999999999999999e-156 < b

   1. Initial program 68.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 67.8

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

   5. Applied rewrites 67.8%

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

Alternative 6: 52.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 2.2e-156) (log1p (exp a)) (fma (fma 0.125 b 0.5) b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 2.2e-156) {
		tmp = log1p(exp(a));
	} else {
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 2.2e-156)
		tmp = log1p(exp(a));
	else
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 2.2e-156], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[(N[(0.125 * b + 0.5), $MachinePrecision] * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.1999999999999999e-156

   1. Initial program 54.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 51.9

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

   5. Applied rewrites 51.9%

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

   if 2.1999999999999999e-156 < b

   1. Initial program 68.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 95.4%

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

   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 65.2%

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

Alternative 7: 58.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\log \left(1 + e^{a}\right) + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)} \]
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. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

5. Applied rewrites 78.0%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 77.7%

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

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{2}, b, \mathsf{log1p}\left(e^{a}\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 57.4%

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

Alternative 8: 49.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.005208333333333333, 0.125\right), b, 0.5\right), b, \log 2\right) \end{array} \]

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return fma(fma(fma((b * b), -0.005208333333333333, 0.125), b, 0.5), b, log(2.0));
}

Julia

a, b = sort([a, b])
function code(a, b)
	return fma(fma(fma(Float64(b * b), -0.005208333333333333, 0.125), b, 0.5), b, log(2.0))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(N[(N[(b * b), $MachinePrecision] * -0.005208333333333333 + 0.125), $MachinePrecision] * b + 0.5), $MachinePrecision] * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.1%

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

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

   7. lower-log.f64 N/A

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

   8. clear-num N/A

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

   9. flip-+ N/A

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

   10. lift-+.f64 N/A

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

   11. lower-/.f64 57.1

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 57.1

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

4. Applied rewrites 57.1%

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

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 53.5

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

7. Applied rewrites 53.5%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 52.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.005208333333333333, 0.125\right), b, 0.5\right), \color{blue}{b}, \log 2\right) \]

11. Final simplification 52.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.005208333333333333, 0.125\right), b, 0.5\right), b, \log 2\right) \]
12. Add Preprocessing

Alternative 9: 50.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right) \end{array} \]

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return fma(fma(0.125, b, 0.5), b, log(2.0));
}

Julia

a, b = sort([a, b])
function code(a, b)
	return fma(fma(0.125, b, 0.5), b, log(2.0))
end

Wolfram

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\log \left(1 + e^{a}\right) + b \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)} \]
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. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

5. Applied rewrites 78.0%

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

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 52.4%

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

Alternative 10: 50.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

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

   7. lower-log.f64 N/A

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

   8. clear-num N/A

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

   9. flip-+ N/A

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

   10. lift-+.f64 N/A

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

   11. lower-/.f64 57.1

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 57.1

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

4. Applied rewrites 57.1%

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

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 53.5

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

7. Applied rewrites 53.5%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 52.3%

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

11. Final simplification 52.3%

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

Alternative 11: 49.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

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

   7. lower-log.f64 N/A

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

   8. clear-num N/A

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

   9. flip-+ N/A

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

   10. lift-+.f64 N/A

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

   11. lower-/.f64 57.1

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 57.1

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

4. Applied rewrites 57.1%

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

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 53.5

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

7. Applied rewrites 53.5%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 51.5%

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

11. Final simplification 51.5%

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

Alternative 12: 49.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 53.8

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

5. Applied rewrites 53.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 51.8%

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

Reproduce

herbie shell --seed 2024318 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))