symmetry log of sum of exp

Percentage Accurate: 54.1% → 99.0%

Time: 19.0s

Alternatives: 12

Speedup: 2.9×

• could not determine a ground truth

  1. a = -837.8396109185207
  2. b = 3.761781707181494e+198

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: 54.1% 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: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := e^{b} + e^{a}\\ \mathbf{if}\;t\_0 \leq 1.5:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;t\_0 \leq 50000000000000:\\ \;\;\;\;\log t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{e^{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp b) (exp a))))
   (if (<= t_0 1.5)
     (/ b (+ (exp a) 1.0))
     (if (<= t_0 50000000000000.0) (log t_0) (/ b (exp (* 2.0 a)))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = exp(b) + exp(a);
	double tmp;
	if (t_0 <= 1.5) {
		tmp = b / (exp(a) + 1.0);
	} else if (t_0 <= 50000000000000.0) {
		tmp = log(t_0);
	} else {
		tmp = b / exp((2.0 * 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) :: t_0
    real(8) :: tmp
    t_0 = exp(b) + exp(a)
    if (t_0 <= 1.5d0) then
        tmp = b / (exp(a) + 1.0d0)
    else if (t_0 <= 50000000000000.0d0) then
        tmp = log(t_0)
    else
        tmp = b / exp((2.0d0 * a))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.exp(b) + Math.exp(a);
	double tmp;
	if (t_0 <= 1.5) {
		tmp = b / (Math.exp(a) + 1.0);
	} else if (t_0 <= 50000000000000.0) {
		tmp = Math.log(t_0);
	} else {
		tmp = b / Math.exp((2.0 * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.exp(b) + math.exp(a)
	tmp = 0
	if t_0 <= 1.5:
		tmp = b / (math.exp(a) + 1.0)
	elif t_0 <= 50000000000000.0:
		tmp = math.log(t_0)
	else:
		tmp = b / math.exp((2.0 * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(exp(b) + exp(a))
	tmp = 0.0
	if (t_0 <= 1.5)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	elseif (t_0 <= 50000000000000.0)
		tmp = log(t_0);
	else
		tmp = Float64(b / exp(Float64(2.0 * a)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = exp(b) + exp(a);
	tmp = 0.0;
	if (t_0 <= 1.5)
		tmp = b / (exp(a) + 1.0);
	elseif (t_0 <= 50000000000000.0)
		tmp = log(t_0);
	else
		tmp = b / exp((2.0 * a));
	end
	tmp_2 = tmp;
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[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.5], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 50000000000000.0], N[Log[t$95$0], $MachinePrecision], N[(b / N[Exp[N[(2.0 * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (exp.f64 a) (exp.f64 b)) < 1.5

   1. Initial program 7.6%

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

   3. Taylor expanded in b around 0 46.7%

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

      1. log1p-def 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in b around inf 46.7%

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

   if 1.5 < (+.f64 (exp.f64 a) (exp.f64 b)) < 5e13

   1. Initial program 99.9%

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

   if 5e13 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 3.2%

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

   3. Taylor expanded in b around 0 12.5%

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

      1. log1p-def 12.5%

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

   5. Simplified 12.5%

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

      1. expm1-log1p-u 12.5%

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

   7. Applied egg-rr 12.5%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 12.5%

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

      2. add-cube-cbrt 12.5%

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

      3. times-frac 12.5%

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

      4. pow2 12.5%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{1}{\color{blue}{{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}\right)}^{2}}} \cdot \frac{b}{\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}} \]

      5. expm1-log1p-u 12.5%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{1}{{\left(\sqrt[3]{\color{blue}{1 + e^{a}}}\right)}^{2}} \cdot \frac{b}{\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}} \]

      6. expm1-log1p-u 12.5%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{1}{{\left(\sqrt[3]{1 + e^{a}}\right)}^{2}} \cdot \frac{b}{\sqrt[3]{\color{blue}{1 + e^{a}}}} \]

   9. Applied egg-rr 12.5%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{\frac{1}{{\left(\sqrt[3]{1 + e^{a}}\right)}^{2}} \cdot \frac{b}{\sqrt[3]{1 + e^{a}}}} \]

   11. Simplified 24.6%

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

   12. Taylor expanded in b around inf 61.2%

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

       1. unpow2 61.2%

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

       2. prod-exp 61.2%

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

       3. count-2 61.2%

          \[\leadsto \frac{b}{e^{\color{blue}{2 \cdot a}}} \]

   14. Simplified 61.2%

       \[\leadsto \color{blue}{\frac{b}{e^{2 \cdot a}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} + e^{a} \leq 1.5:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{b} + e^{a} \leq 50000000000000:\\ \;\;\;\;\log \left(e^{b} + e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{e^{2 \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 2:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + \frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{e^{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 2.0)
   (+ (log1p (exp a)) (/ b (+ (exp a) 1.0)))
   (/ b (exp (* 2.0 a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(b) <= 2.0) {
		tmp = log1p(exp(a)) + (b / (exp(a) + 1.0));
	} else {
		tmp = b / exp((2.0 * a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(b) <= 2.0) {
		tmp = Math.log1p(Math.exp(a)) + (b / (Math.exp(a) + 1.0));
	} else {
		tmp = b / Math.exp((2.0 * a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(b) <= 2.0:
		tmp = math.log1p(math.exp(a)) + (b / (math.exp(a) + 1.0))
	else:
		tmp = b / math.exp((2.0 * a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(b) <= 2.0)
		tmp = Float64(log1p(exp(a)) + Float64(b / Float64(exp(a) + 1.0)));
	else
		tmp = Float64(b / exp(Float64(2.0 * a)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[b], $MachinePrecision], 2.0], N[(N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision] + N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / N[Exp[N[(2.0 * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 2

   1. Initial program 52.5%

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

   3. Taylor expanded in b around 0 71.7%

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

      1. log1p-def 72.1%

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

   5. Simplified 72.1%

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

   if 2 < (exp.f64 b)

   1. Initial program 3.2%

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

   3. Taylor expanded in b around 0 18.8%

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

      1. log1p-def 18.8%

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

   5. Simplified 18.8%

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

      1. expm1-log1p-u 18.8%

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

   7. Applied egg-rr 18.8%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 18.8%

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

      2. add-cube-cbrt 18.8%

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

      3. times-frac 18.8%

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

      4. pow2 18.8%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{1}{\color{blue}{{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}\right)}^{2}}} \cdot \frac{b}{\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}} \]

      5. expm1-log1p-u 18.8%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{1}{{\left(\sqrt[3]{\color{blue}{1 + e^{a}}}\right)}^{2}} \cdot \frac{b}{\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}} \]

      6. expm1-log1p-u 18.8%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{1}{{\left(\sqrt[3]{1 + e^{a}}\right)}^{2}} \cdot \frac{b}{\sqrt[3]{\color{blue}{1 + e^{a}}}} \]

   9. Applied egg-rr 18.8%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{\frac{1}{{\left(\sqrt[3]{1 + e^{a}}\right)}^{2}} \cdot \frac{b}{\sqrt[3]{1 + e^{a}}}} \]

   11. Simplified 38.8%

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

   12. Taylor expanded in b around inf 100.0%

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

       1. unpow2 100.0%

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

       2. prod-exp 100.0%

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

       3. count-2 100.0%

          \[\leadsto \frac{b}{e^{\color{blue}{2 \cdot a}}} \]

   14. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{b}{e^{2 \cdot a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 2:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + \frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{e^{2 \cdot a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-71}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-71) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (+ b 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-71) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((exp(a) + (b + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 5d-71) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + (b + 1.0d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-71) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((Math.exp(a) + (b + 1.0)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-71:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((math.exp(a) + (b + 1.0)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-71)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(exp(a) + Float64(b + 1.0)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-71)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((exp(a) + (b + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-71], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(b + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999998e-71

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0 96.9%

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

      1. log1p-def 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in b around inf 96.9%

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

   if 4.99999999999999998e-71 < (exp.f64 a)

   1. Initial program 65.4%

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

   3. Taylor expanded in b around 0 62.1%

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

      1. associate-+r+ 62.1%

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

      2. +-commutative 62.1%

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

   5. Simplified 62.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-71}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around inf 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.1%

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

   3. Taylor expanded in b around 0 62.4%

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

      1. log1p-def 62.9%

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

   5. Simplified 62.9%

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

4. Final simplification 71.2%

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

Alternative 5: 97.4% 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 8.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around inf 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0 62.8%

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

      1. log1p-def 62.8%

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

   5. Simplified 62.8%

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

4. Final simplification 71.1%

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

Alternative 6: 96.8% accurate, 1.4× speedup

Math

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

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 8.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around inf 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.1%

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

   3. Taylor expanded in b around 0 62.7%

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

      1. log1p-def 63.2%

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

   5. Simplified 63.2%

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

   6. Taylor expanded in a around 0 61.3%

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

      1. +-commutative 61.3%

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

   8. Simplified 61.3%

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

4. Final simplification 70.0%

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

Alternative 7: 56.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -175:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -175.0) (/ b 2.0) (+ (* b 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -175.0) {
		tmp = b / 2.0;
	} 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 (a <= (-175.0d0)) then
        tmp = b / 2.0d0
    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 (a <= -175.0) {
		tmp = b / 2.0;
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -175.0:
		tmp = b / 2.0
	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 (a <= -175.0)
		tmp = Float64(b / 2.0);
	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 (a <= -175.0)
		tmp = b / 2.0;
	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[a, -175.0], N[(b / 2.0), $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -175

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around inf 98.4%

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

   7. Taylor expanded in a around 0 18.5%

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

   if -175 < a

   1. Initial program 65.1%

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

   3. Taylor expanded in b around 0 62.7%

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

      1. log1p-def 63.2%

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

   5. Simplified 63.2%

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

   6. Taylor expanded in a around 0 61.3%

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

      1. +-commutative 61.3%

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

   8. Simplified 61.3%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -175:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b / 2.0;
	} else {
		tmp = log((2.0 + 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 (a <= (-1.0d0)) then
        tmp = b / 2.0d0
    else
        tmp = log((2.0d0 + a))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.0)
		tmp = b / 2.0;
	else
		tmp = log((2.0 + 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[a, -1.0], N[(b / 2.0), $MachinePrecision], N[Log[N[(2.0 + a), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0 96.9%

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

      1. log1p-def 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in b around inf 96.9%

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

   7. Taylor expanded in a around 0 18.3%

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

   if -1 < a

   1. Initial program 65.4%

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

   3. Taylor expanded in b around 0 62.7%

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

   4. Taylor expanded in a around 0 62.1%

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

      1. +-commutative 62.1%

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

   6. Simplified 62.1%

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

4. Final simplification 51.6%

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

Alternative 9: 56.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -170:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -170.0) (/ b 2.0) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -170.0) {
		tmp = b / 2.0;
	} 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 <= (-170.0d0)) then
        tmp = b / 2.0d0
    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 <= -170.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -170.0:
		tmp = b / 2.0
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -170.0)
		tmp = Float64(b / 2.0);
	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 <= -170.0)
		tmp = b / 2.0;
	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, -170.0], N[(b / 2.0), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -170

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around inf 98.4%

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

   7. Taylor expanded in a around 0 18.5%

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

   if -170 < a

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0 62.8%

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

   4. Taylor expanded in b around 0 60.4%

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

      1. +-commutative 60.4%

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

   6. Simplified 60.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -170:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -190:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;b + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -190.0) (/ b 2.0) (+ b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -190.0) {
		tmp = b / 2.0;
	} else {
		tmp = b + 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 <= (-190.0d0)) then
        tmp = b / 2.0d0
    else
        tmp = b + 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 <= -190.0) {
		tmp = b / 2.0;
	} else {
		tmp = b + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -190.0:
		tmp = b / 2.0
	else:
		tmp = b + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -190.0)
		tmp = Float64(b / 2.0);
	else
		tmp = Float64(b + 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 <= -190.0)
		tmp = b / 2.0;
	else
		tmp = b + 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, -190.0], N[(b / 2.0), $MachinePrecision], N[(b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -190

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around inf 98.4%

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

   7. Taylor expanded in a around 0 18.5%

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

   if -190 < a

   1. Initial program 65.1%

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

   3. Taylor expanded in b around 0 62.7%

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

      1. log1p-def 63.2%

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

   5. Simplified 63.2%

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

      1. expm1-log1p-u 63.2%

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

   7. Applied egg-rr 63.2%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 63.2%

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

      2. add-cube-cbrt 63.2%

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

      3. times-frac 63.2%

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

      4. pow2 63.2%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{1}{\color{blue}{{\left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}\right)}^{2}}} \cdot \frac{b}{\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}} \]

      5. expm1-log1p-u 63.2%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{1}{{\left(\sqrt[3]{\color{blue}{1 + e^{a}}}\right)}^{2}} \cdot \frac{b}{\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}} \]

      6. expm1-log1p-u 63.2%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{1}{{\left(\sqrt[3]{1 + e^{a}}\right)}^{2}} \cdot \frac{b}{\sqrt[3]{\color{blue}{1 + e^{a}}}} \]

   9. Applied egg-rr 63.2%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{\frac{1}{{\left(\sqrt[3]{1 + e^{a}}\right)}^{2}} \cdot \frac{b}{\sqrt[3]{1 + e^{a}}}} \]

   11. Simplified 62.3%

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

   12. Taylor expanded in a around 0 60.9%

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

       1. +-commutative 60.9%

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

   14. Simplified 60.9%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -190:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;b + \log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -165:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -165.0) (/ b 2.0) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -165.0) {
		tmp = b / 2.0;
	} 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 <= (-165.0d0)) then
        tmp = b / 2.0d0
    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 <= -165.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -165.0:
		tmp = b / 2.0
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -165.0)
		tmp = Float64(b / 2.0);
	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 <= -165.0)
		tmp = b / 2.0;
	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, -165.0], N[(b / 2.0), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -165

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in b around inf 98.4%

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

   7. Taylor expanded in a around 0 18.5%

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

   if -165 < a

   1. Initial program 65.1%

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

   3. Taylor expanded in b around 0 62.4%

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

      1. log1p-def 62.9%

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

   5. Simplified 62.9%

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

   6. Taylor expanded in a around 0 61.0%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -165:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 11.9% accurate, 101.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{b}{2} \end{array} \]

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return b / 2.0;
}

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 / 2.0d0
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b / 2.0;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b / 2.0

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b / 2.0)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = b / 2.0;
end

Wolfram

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

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in b around 0 71.1%

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

   1. log1p-def 71.5%

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

5. Simplified 71.5%

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

6. Taylor expanded in b around inf 25.9%

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

7. Taylor expanded in a around 0 7.2%

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

8. Final simplification 7.2%

   \[\leadsto \frac{b}{2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))