symmetry log of sum of exp

Percentage Accurate: 53.5% → 98.9%

Time: 18.7s

Alternatives: 11

Speedup: 2.9×

• could not determine a ground truth

  1. a = 4.970187881437105e+51
  2. b = -8.663164345392522e+55

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: 53.5% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + exp(b)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.5%

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

      1. add-sqr-sqrt 8.4%

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

      2. log-prod 8.4%

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

   3. Applied egg-rr 8.4%

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

      1. log-prod 8.4%

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

      2. rem-square-sqrt 8.5%

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

      3. log1p-expm1 8.5%

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

      4. expm1-def 8.5%

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

      5. rem-exp-log 8.5%

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

      6. associate--l+ 8.5%

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

      7. expm1-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 61.6%

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

4. Final simplification 70.5%

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

Alternative 2: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.5%

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

      1. add-sqr-sqrt 8.4%

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

      2. log-prod 8.4%

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

   3. Applied egg-rr 8.4%

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

      1. log-prod 8.4%

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

      2. rem-square-sqrt 8.5%

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

      3. log1p-expm1 8.5%

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

      4. expm1-def 8.5%

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

      5. rem-exp-log 8.5%

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

      6. associate--l+ 8.5%

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

      7. expm1-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 61.6%

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

   2. Taylor expanded in a around 0 58.5%

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

      1. log1p-def 58.5%

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

   4. Simplified 58.5%

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

4. Final simplification 68.1%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (log1p (+ (exp a) (expm1 b))))

C

assert(a < b);
double code(double a, double b) {
	return log1p((exp(a) + expm1(b)));
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log1p((Math.exp(a) + Math.expm1(b)));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p((math.exp(a) + math.expm1(b)))

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(Float64(exp(a) + expm1(b)))
end

Wolfram

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

Derivation

1. Initial program 49.4%

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

   1. add-sqr-sqrt 48.4%

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

   2. log-prod 48.7%

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

3. Applied egg-rr 48.7%

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

   1. log-prod 48.4%

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

   2. rem-square-sqrt 49.4%

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

   3. log1p-expm1 49.4%

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

   4. expm1-def 49.4%

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

   5. rem-exp-log 49.4%

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

   6. associate--l+ 49.8%

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

   7. expm1-def 70.5%

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

5. Simplified 70.5%

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

6. Final simplification 70.5%

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

Alternative 4: 97.8% accurate, 1.4× speedup

Math

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

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 (+ (exp a) 1.0))
   (log (+ b (+ 2.0 (+ a (* 0.5 (+ (* a a) (* b b)))))))))

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((b + (2.0d0 + (a + (0.5d0 * ((a * a) + (b * b)))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((b + (2.0 + (a + (0.5 * ((a * a) + (b * b)))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((b + (2.0 + (a + (0.5 * ((a * a) + (b * 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(exp(a) + 1.0));
	else
		tmp = log(Float64(b + Float64(2.0 + Float64(a + Float64(0.5 * Float64(Float64(a * a) + Float64(b * b)))))));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.5%

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

      1. add-sqr-sqrt 8.4%

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

      2. log-prod 8.4%

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

   3. Applied egg-rr 8.4%

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

      1. log-prod 8.4%

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

      2. rem-square-sqrt 8.5%

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

      3. log1p-expm1 8.5%

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

      4. expm1-def 8.5%

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

      5. rem-exp-log 8.5%

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

      6. associate--l+ 8.5%

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

      7. expm1-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 61.6%

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

   2. Taylor expanded in b around 0 58.8%

      \[\leadsto \log \color{blue}{\left(1 + \left(b + \left(e^{a} + 0.5 \cdot {b}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 58.8%

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

      2. associate-+l+ 58.8%

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

      3. associate-+l+ 58.8%

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

      4. *-commutative 58.8%

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

      5. unpow2 58.8%

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

      6. associate-*l* 58.8%

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

      7. fma-def 58.8%

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

   4. Simplified 58.8%

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

   5. Taylor expanded in a around 0 57.6%

      \[\leadsto \log \left(b + \color{blue}{\left(2 + \left(a + \left(0.5 \cdot {a}^{2} + 0.5 \cdot {b}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 57.6%

         \[\leadsto \log \left(b + \left(2 + \left(a + \left(0.5 \cdot {a}^{2} + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

      2. distribute-lft-out 57.6%

         \[\leadsto \log \left(b + \left(2 + \left(a + \color{blue}{0.5 \cdot \left({a}^{2} + b \cdot b\right)}\right)\right)\right) \]

      3. unpow2 57.6%

         \[\leadsto \log \left(b + \left(2 + \left(a + 0.5 \cdot \left(\color{blue}{a \cdot a} + b \cdot b\right)\right)\right)\right) \]

   7. Simplified 57.6%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(2 + \left(a + 0.5 \cdot \left(a \cdot a + b \cdot b\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 97.6% accurate, 1.4× speedup

Math

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

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 (+ (exp a) 1.0))
   (log (+ b (+ 2.0 (+ a (* 0.5 (* a a))))))))

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((b + (2.0d0 + (a + (0.5d0 * (a * a))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((b + (2.0 + (a + (0.5 * (a * a))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((b + (2.0 + (a + (0.5 * (a * 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(exp(a) + 1.0));
	else
		tmp = log(Float64(b + Float64(2.0 + Float64(a + Float64(0.5 * Float64(a * a))))));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.5%

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

      1. add-sqr-sqrt 8.4%

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

      2. log-prod 8.4%

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

   3. Applied egg-rr 8.4%

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

      1. log-prod 8.4%

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

      2. rem-square-sqrt 8.5%

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

      3. log1p-expm1 8.5%

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

      4. expm1-def 8.5%

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

      5. rem-exp-log 8.5%

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

      6. associate--l+ 8.5%

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

      7. expm1-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 61.6%

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

   2. Taylor expanded in b around 0 58.8%

      \[\leadsto \log \color{blue}{\left(1 + \left(b + \left(e^{a} + 0.5 \cdot {b}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 58.8%

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

      2. associate-+l+ 58.8%

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

      3. associate-+l+ 58.8%

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

      4. *-commutative 58.8%

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

      5. unpow2 58.8%

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

      6. associate-*l* 58.8%

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

      7. fma-def 58.8%

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

   4. Simplified 58.8%

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

   5. Taylor expanded in a around 0 57.6%

      \[\leadsto \log \left(b + \color{blue}{\left(2 + \left(a + \left(0.5 \cdot {a}^{2} + 0.5 \cdot {b}^{2}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 57.6%

         \[\leadsto \log \left(b + \left(2 + \left(a + \left(0.5 \cdot {a}^{2} + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

      2. distribute-lft-out 57.6%

         \[\leadsto \log \left(b + \left(2 + \left(a + \color{blue}{0.5 \cdot \left({a}^{2} + b \cdot b\right)}\right)\right)\right) \]

      3. unpow2 57.6%

         \[\leadsto \log \left(b + \left(2 + \left(a + 0.5 \cdot \left(\color{blue}{a \cdot a} + b \cdot b\right)\right)\right)\right) \]

   7. Simplified 57.6%

      \[\leadsto \log \left(b + \color{blue}{\left(2 + \left(a + 0.5 \cdot \left(a \cdot a + b \cdot b\right)\right)\right)}\right) \]

   8. Taylor expanded in a around inf 56.3%

      \[\leadsto \log \left(b + \left(2 + \color{blue}{\left(a + 0.5 \cdot {a}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 56.3%

         \[\leadsto \log \left(b + \left(2 + \left(a + 0.5 \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)\right) \]

   10. Simplified 56.3%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(2 + \left(a + 0.5 \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 97.0% accurate, 1.4× speedup

Math

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

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 (+ (exp a) 1.0))
   (log (+ b (+ 2.0 (* b (* b 0.5)))))))

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((b + (2.0d0 + (b * (b * 0.5d0)))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((b + (2.0 + (b * (b * 0.5)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((b + (2.0 + (b * (b * 0.5)))))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.5%

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

      1. add-sqr-sqrt 8.4%

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

      2. log-prod 8.4%

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

   3. Applied egg-rr 8.4%

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

      1. log-prod 8.4%

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

      2. rem-square-sqrt 8.5%

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

      3. log1p-expm1 8.5%

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

      4. expm1-def 8.5%

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

      5. rem-exp-log 8.5%

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

      6. associate--l+ 8.5%

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

      7. expm1-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 61.6%

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

   2. Taylor expanded in b around 0 58.8%

      \[\leadsto \log \color{blue}{\left(1 + \left(b + \left(e^{a} + 0.5 \cdot {b}^{2}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 58.8%

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

      2. associate-+l+ 58.8%

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

      3. associate-+l+ 58.8%

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

      4. *-commutative 58.8%

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

      5. unpow2 58.8%

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

      6. associate-*l* 58.8%

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

      7. fma-def 58.8%

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

   4. Simplified 58.8%

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

   5. Taylor expanded in a around 0 57.0%

      \[\leadsto \log \left(b + \color{blue}{\left(2 + 0.5 \cdot {b}^{2}\right)}\right) \]
   6. Step-by-step derivation

      1. unpow2 57.0%

         \[\leadsto \log \left(b + \left(2 + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

      2. *-commutative 57.0%

         \[\leadsto \log \left(b + \left(2 + \color{blue}{\left(b \cdot b\right) \cdot 0.5}\right)\right) \]

      3. associate-*r* 57.0%

         \[\leadsto \log \left(b + \left(2 + \color{blue}{b \cdot \left(b \cdot 0.5\right)}\right)\right) \]

   7. Simplified 57.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(2 + b \cdot \left(b \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 7: 96.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = (b * 0.5d0) + log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.5%

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

      1. add-sqr-sqrt 8.4%

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

      2. log-prod 8.4%

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

   3. Applied egg-rr 8.4%

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

      1. log-prod 8.4%

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

      2. rem-square-sqrt 8.5%

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

      3. log1p-expm1 8.5%

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

      4. expm1-def 8.5%

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

      5. rem-exp-log 8.5%

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

      6. associate--l+ 8.5%

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

      7. expm1-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 61.6%

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

   2. Taylor expanded in a around 0 58.5%

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

   3. Taylor expanded in b around 0 56.7%

      \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
   4. Step-by-step derivation

      1. *-commutative 56.7%

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

   5. Simplified 56.7%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 8: 56.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -115:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -115.0) (* b 0.5) (+ (* b 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -115.0) {
		tmp = b * 0.5;
	} else {
		tmp = (b * 0.5) + log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-115.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = (b * 0.5d0) + log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -115.0) {
		tmp = b * 0.5;
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -115.0:
		tmp = b * 0.5
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -115.0)
		tmp = Float64(b * 0.5);
	else
		tmp = Float64(Float64(b * 0.5) + log(2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -115.0)
		tmp = b * 0.5;
	else
		tmp = (b * 0.5) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -115.0], N[(b * 0.5), $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -115

   1. Initial program 8.5%

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

   2. Taylor expanded in a around 0 3.6%

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

   3. Taylor expanded in b around 0 3.9%

      \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
   4. Step-by-step derivation

      1. *-commutative 3.9%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in b around inf 18.8%

      \[\leadsto \color{blue}{0.5 \cdot b} \]
   7. Step-by-step derivation

      1. *-commutative 18.8%

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

   8. Simplified 18.8%

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

   if -115 < a

   1. Initial program 61.6%

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

   2. Taylor expanded in a around 0 58.5%

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

   3. Taylor expanded in b around 0 56.7%

      \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
   4. Step-by-step derivation

      1. *-commutative 56.7%

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

   5. Simplified 56.7%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -115:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 9: 56.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -100:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -100.0) (* b 0.5) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -100.0) {
		tmp = b * 0.5;
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-100.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = log((b + 2.0d0))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -100.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -100.0:
		tmp = b * 0.5
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -100.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log(Float64(b + 2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -100.0)
		tmp = b * 0.5;
	else
		tmp = log((b + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -100.0], N[(b * 0.5), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -100

   1. Initial program 8.5%

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

   2. Taylor expanded in a around 0 3.6%

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

   3. Taylor expanded in b around 0 3.9%

      \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
   4. Step-by-step derivation

      1. *-commutative 3.9%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in b around inf 18.8%

      \[\leadsto \color{blue}{0.5 \cdot b} \]
   7. Step-by-step derivation

      1. *-commutative 18.8%

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

   8. Simplified 18.8%

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

   if -100 < a

   1. Initial program 61.6%

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

   2. Taylor expanded in a around 0 58.5%

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

   3. Taylor expanded in b around 0 55.6%

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

      1. +-commutative 55.6%

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

   5. Simplified 55.6%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -100:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 10: 55.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -80:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -80.0) (* b 0.5) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -80.0) {
		tmp = b * 0.5;
	} else {
		tmp = log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-80.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -80.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -80.0:
		tmp = b * 0.5
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -80.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -80.0)
		tmp = b * 0.5;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -80.0], N[(b * 0.5), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -80

   1. Initial program 8.5%

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

   2. Taylor expanded in a around 0 3.6%

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

   3. Taylor expanded in b around 0 3.9%

      \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
   4. Step-by-step derivation

      1. *-commutative 3.9%

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

   5. Simplified 3.9%

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

   6. Taylor expanded in b around inf 18.8%

      \[\leadsto \color{blue}{0.5 \cdot b} \]
   7. Step-by-step derivation

      1. *-commutative 18.8%

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

   8. Simplified 18.8%

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

   if -80 < a

   1. Initial program 61.6%

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

   2. Taylor expanded in b around 0 58.3%

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

      1. log1p-def 58.9%

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

   4. Simplified 58.9%

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

   5. Taylor expanded in a around 0 56.4%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -80:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 11: 12.0% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

2. Taylor expanded in a around 0 45.8%

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

3. Taylor expanded in b around 0 44.5%

   \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
4. Step-by-step derivation

   1. *-commutative 44.5%

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

5. Simplified 44.5%

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

6. Taylor expanded in b around inf 6.9%

   \[\leadsto \color{blue}{0.5 \cdot b} \]
7. Step-by-step derivation

   1. *-commutative 6.9%

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

8. Simplified 6.9%

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

9. Final simplification 6.9%

   \[\leadsto b \cdot 0.5 \]

Reproduce

herbie shell --seed 2023283 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))