symmetry log of sum of exp

Percentage Accurate: 53.3% → 98.1%

Time: 9.0s

Alternatives: 9

Speedup: 2.8×

• could not determine a ground truth

  1. a = 2.3333751812608804e+51
  2. b = 9.89099374239273e+96

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in b around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-exp.f64 71.7

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

5. Applied rewrites 71.7%

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

7. Applied rewrites 71.7%

   \[\leadsto \frac{b \cdot \mathsf{expm1}\left(a \cdot 3\right)}{\mathsf{expm1}\left(a + a\right) \cdot \mathsf{fma}\left(e^{a}, e^{a}, 1 + e^{a}\right)} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

9. Applied rewrites 71.9%

   \[\leadsto \frac{\mathsf{expm1}\left(a\right) \cdot b}{\mathsf{expm1}\left(a\right)} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

10. Taylor expanded in a around inf

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

12. Applied rewrites 71.9%

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

Alternative 2: 55.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (exp.f64 a) (exp.f64 b))) < 0.5

   1. Initial program 9.5%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 5.6

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

   5. Applied rewrites 5.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 2.9%

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

   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 9.8%

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

   if 0.5 < (log.f64 (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 97.6%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.3%

      \[\leadsto \frac{b \cdot \mathsf{expm1}\left(a \cdot 3\right)}{\mathsf{expm1}\left(a + a\right) \cdot \mathsf{fma}\left(e^{a}, e^{a}, 1 + e^{a}\right)} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   9. Applied rewrites 97.7%

      \[\leadsto \frac{\mathsf{expm1}\left(a\right) \cdot b}{\mathsf{expm1}\left(a\right)} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   10. Taylor expanded in a around 0

       \[\leadsto b + \color{blue}{\log 2} \]
   11. Step-by-step derivation

   12. Applied rewrites 95.2%

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

Alternative 3: 56.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -640:\\ \;\;\;\;\mathsf{fma}\left(0.125, b, 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -640.0) (* (fma 0.125 b 0.5) b) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -640.0) {
		tmp = fma(0.125, b, 0.5) * b;
	} else {
		tmp = log1p(exp(a));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -640.0)
		tmp = Float64(fma(0.125, b, 0.5) * b);
	else
		tmp = log1p(exp(a));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -640.0], N[(N[(0.125 * b + 0.5), $MachinePrecision] * b), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -640

   1. Initial program 8.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 3.8

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 3.9%

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

   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%

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

   if -640 < a

   1. Initial program 66.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 63.3

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

   5. Applied rewrites 63.3%

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

Alternative 4: 56.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.6) {
		tmp = fma(0.125, b, 0.5) * b;
	} else {
		tmp = fma(fma(fma((a * a), -0.005208333333333333, 0.125), a, 0.5), a, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.6)
		tmp = Float64(fma(0.125, b, 0.5) * b);
	else
		tmp = fma(fma(fma(Float64(a * a), -0.005208333333333333, 0.125), a, 0.5), a, log(2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -2.6], N[(N[(0.125 * b + 0.5), $MachinePrecision] * b), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * -0.005208333333333333 + 0.125), $MachinePrecision] * a + 0.5), $MachinePrecision] * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.60000000000000009

   1. Initial program 8.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 3.8

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 3.9%

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

   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%

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

   if -2.60000000000000009 < a

   1. Initial program 66.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 62.7%

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

Alternative 5: 55.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -120:\\ \;\;\;\;\mathsf{fma}\left(0.125, b, 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -120.0) (* (fma 0.125 b 0.5) b) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -120.0) {
		tmp = fma(0.125, b, 0.5) * b;
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -120.0)
		tmp = Float64(fma(0.125, b, 0.5) * b);
	else
		tmp = log1p(1.0);
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -120.0], N[(N[(0.125 * b + 0.5), $MachinePrecision] * b), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -120

   1. Initial program 8.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 3.8

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 3.9%

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

   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%

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

   if -120 < a

   1. Initial program 66.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 62.0%

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

Alternative 6: 11.9% accurate, 25.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 52.2%

   \[\log \left(e^{a} + e^{b}\right) \]
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 49.2

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 47.3%

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

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 6.9%

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

Alternative 7: 5.3% accurate, 27.6× speedup

Math

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

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(Float64(b * 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[(N[(b * b), $MachinePrecision] * 0.125), $MachinePrecision]

Derivation

1. Initial program 52.2%

   \[\log \left(e^{a} + e^{b}\right) \]
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 49.2

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 47.3%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 4.0%

    \[\leadsto \left(b \cdot b\right) \cdot 0.125 \]
12. Add Preprocessing

Alternative 8: 3.2% accurate, 27.6× speedup

Math

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

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(Float64(a * 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[(N[(a * a), $MachinePrecision] * 0.125), $MachinePrecision]

Derivation

1. Initial program 52.2%

   \[\log \left(e^{a} + e^{b}\right) \]
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 49.0

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

5. Applied rewrites 49.0%

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

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 47.8%

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

9. Taylor expanded in a around inf

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

11. Applied rewrites 4.3%

    \[\leadsto \left(a \cdot a\right) \cdot 0.125 \]
12. Add Preprocessing

Alternative 9: 2.6% accurate, 50.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

   \[\log \left(e^{a} + e^{b}\right) \]
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 49.0

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

5. Applied rewrites 49.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 47.8%

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

9. Taylor expanded in a around inf

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

11. Applied rewrites 7.6%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))