symmetry log of sum of exp

Percentage Accurate: 54.7% → 98.8%

Time: 8.0s

Alternatives: 13

Speedup: 2.8×

• could not determine a ground truth

  1. a = -2.261103164764831e-229
  2. b = 9.496636557452436e+144

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -37.0) {
		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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-37.0d0)) 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 (a <= -37.0) {
		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 a <= -37.0:
		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 (a <= -37.0)
		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 (a <= -37.0)
		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[a, -37.0], 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 a < -37

   1. Initial program 8.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. lift-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lift-exp.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -37 < a

   1. Initial program 67.0%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := e^{a} - -1\\ t_1 := {t\_0}^{-1}\\ \mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot b, t\_1 - {t\_0}^{-2}, t\_1\right), b, \mathsf{log1p}\left(e^{a}\right)\right) \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (exp a) -1.0)) (t_1 (pow t_0 -1.0)))
   (fma (fma (* 0.5 b) (- t_1 (pow t_0 -2.0)) t_1) b (log1p (exp a)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = exp(a) - -1.0;
	double t_1 = pow(t_0, -1.0);
	return fma(fma((0.5 * b), (t_1 - pow(t_0, -2.0)), t_1), b, log1p(exp(a)));
}

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(exp(a) - -1.0)
	t_1 = t_0 ^ -1.0
	return fma(fma(Float64(0.5 * b), Float64(t_1 - (t_0 ^ -2.0)), t_1), b, log1p(exp(a)))
end

Wolfram

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

Derivation

1. Initial program 50.7%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 74.1%

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

6. Final simplification 74.1%

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

Alternative 3: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{a} + e^{b}\right) \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \mathsf{log1p}\left(e^{a}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (log (+ (exp a) (exp b))) 4e-8)
   (/ b (+ (exp a) 1.0))
   (fma (fma 0.125 b 0.5) b (log1p (exp a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (log((exp(a) + exp(b))) <= 4e-8) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = fma(fma(0.125, b, 0.5), b, log1p(exp(a)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (log(Float64(exp(a) + exp(b))) <= 4e-8)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = fma(fma(0.125, b, 0.5), b, 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[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4e-8], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * b + 0.5), $MachinePrecision] * b + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (exp.f64 a) (exp.f64 b))) < 4.0000000000000001e-8

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in b around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. lift-exp.f64 52.9

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lift-exp.f64 52.9

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

   11. Applied rewrites 52.9%

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

   if 4.0000000000000001e-8 < (log.f64 (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{8} \cdot b + \frac{1}{2}, b, \mathsf{log1p}\left(e^{a}\right)\right) \]

      2. lower-fma.f64 98.4

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

   8. Applied rewrites 98.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{a} + e^{b}\right) \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \mathsf{log1p}\left(e^{a}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{a} + e^{b}\right) \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (log (+ (exp a) (exp b))) 4e-8)
   (/ b (+ (exp a) 1.0))
   (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (log((exp(a) + exp(b))) <= 4e-8) {
		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.log((Math.exp(a) + Math.exp(b))) <= 4e-8) {
		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.log((math.exp(a) + math.exp(b))) <= 4e-8:
		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 (log(Float64(exp(a) + exp(b))) <= 4e-8)
		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[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4e-8], 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 (log.f64 (+.f64 (exp.f64 a) (exp.f64 b))) < 4.0000000000000001e-8

   1. Initial program 9.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in b around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. lift-exp.f64 52.9

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

   8. Applied rewrites 52.9%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lift-exp.f64 52.9

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

   11. Applied rewrites 52.9%

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

   if 4.0000000000000001e-8 < (log.f64 (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lift-exp.f64 95.0

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

   5. Applied rewrites 95.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{a} + e^{b}\right) \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.7%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower--.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lift-exp.f64 73.7

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

5. Applied rewrites 73.7%

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

6. Final simplification 73.7%

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

Alternative 6: 97.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -36:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -36.0) (/ b (+ (exp a) 1.0)) (fma (fma 0.125 b 0.5) b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -36.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -36.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -36.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * b + 0.5), $MachinePrecision] * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -36

   1. Initial program 8.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. lift-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lift-exp.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -36 < a

   1. Initial program 67.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right) + \log 2 \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{8} \cdot b\right) \cdot b + \log 2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{8} \cdot b, b, \log 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{8} \cdot b + \frac{1}{2}, b, \log 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, b, \frac{1}{2}\right), b, \log 2\right) \]

      6. lower-log.f64 62.0

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

   8. Applied rewrites 62.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -36:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -63:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -63.0) (* 0.5 b) (fma (fma 0.125 b 0.5) b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -63.0) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -63.0)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -63.0], N[(0.5 * b), $MachinePrecision], N[(N[(0.125 * b + 0.5), $MachinePrecision] * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -63

   1. Initial program 8.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. lift-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lift-exp.f64 100.0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in a around 0

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

       1. lower-*.f64 18.8

          \[\leadsto 0.5 \cdot b \]

   14. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -63 < a

   1. Initial program 67.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right) + \log 2 \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} + \frac{1}{8} \cdot b\right) \cdot b + \log 2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{8} \cdot b, b, \log 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{8} \cdot b + \frac{1}{2}, b, \log 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, b, \frac{1}{2}\right), b, \log 2\right) \]

      6. lower-log.f64 62.0

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

   8. Applied rewrites 62.0%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -63:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -130:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -130.0) (* 0.5 b) (fma 0.5 b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -130.0) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(0.5, b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -130.0)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(0.5, b, log(2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -130.0], N[(0.5 * b), $MachinePrecision], N[(0.5 * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -130

   1. Initial program 8.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. lift-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lift-exp.f64 100.0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in a around 0

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

       1. lower-*.f64 18.8

          \[\leadsto 0.5 \cdot b \]

   14. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -130 < a

   1. Initial program 67.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in b around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. lift-exp.f64 63.5

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

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot b + \log 2 \]

       2. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, b, \log 2\right) \]

       3. lift-log.f64 61.4

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

   11. Applied rewrites 61.4%

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

Alternative 9: 57.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, a, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.4) (* 0.5 b) (fma 0.5 a (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.4) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(0.5, a, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.4)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(0.5, a, log(2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.4], N[(0.5 * b), $MachinePrecision], N[(0.5 * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3999999999999999

   1. Initial program 8.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. lift-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lift-exp.f64 100.0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in a around 0

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

       1. lower-*.f64 18.8

          \[\leadsto 0.5 \cdot b \]

   14. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -1.3999999999999999 < a

   1. Initial program 67.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lift-exp.f64 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot a + \log 2 \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, a, \log 2\right) \]

      3. lower-log.f64 62.1

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

   8. Applied rewrites 62.1%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, a, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log1p((1.0 + a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.0:
		tmp = 0.5 * b
	else:
		tmp = math.log1p((1.0 + a))
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 8.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. lift-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lift-exp.f64 100.0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in a around 0

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

       1. lower-*.f64 18.8

          \[\leadsto 0.5 \cdot b \]

   14. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -1 < a

   1. Initial program 67.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lift-exp.f64 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in a around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. lower-+.f64 62.0

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

   8. Applied rewrites 62.0%

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

4. Final simplification 50.0%

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

Alternative 11: 56.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -63:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -63.0) (* 0.5 b) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -63.0) {
		tmp = 0.5 * b;
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -63.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log1p(1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -63.0:
		tmp = 0.5 * b
	else:
		tmp = math.log1p(1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -63.0)
		tmp = Float64(0.5 * b);
	else
		tmp = log1p(1.0);
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -63.0], N[(0.5 * b), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -63

   1. Initial program 8.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. sinh-+-cosh-rev N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-log1p.f64 N/A

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

      8. lift-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lift-exp.f64 100.0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in a around 0

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

       1. lower-*.f64 18.8

          \[\leadsto 0.5 \cdot b \]

   14. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -63 < a

   1. Initial program 67.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lift-exp.f64 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in a around 0

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

      1. sinh-+-cosh-rev 61.3

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

   8. Applied rewrites 61.3%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -63:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 11.8% accurate, 50.7× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0 * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.7%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 74.1%

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

6. Taylor expanded in b around 0

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

   1. sinh-+-cosh-rev N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-log1p.f64 N/A

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

   8. lift-exp.f64 73.7

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

8. Applied rewrites 73.7%

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

9. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 N/A

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

    4. lift-exp.f64 30.2

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

11. Applied rewrites 30.2%

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

12. Taylor expanded in a around 0

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

    1. lower-*.f64 7.6

       \[\leadsto 0.5 \cdot b \]

14. Applied rewrites 7.6%

    \[\leadsto 0.5 \cdot b \]
15. Add Preprocessing

Alternative 13: 2.6% accurate, 50.7× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0 * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.7%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lift-exp.f64 47.5

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

5. Applied rewrites 47.5%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot a + \log 2 \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, a, \log 2\right) \]

   3. lower-log.f64 45.6

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

8. Applied rewrites 45.6%

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

9. Taylor expanded in a around inf

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

    1. lower-*.f64 7.5

       \[\leadsto 0.5 \cdot a \]

11. Applied rewrites 7.5%

    \[\leadsto 0.5 \cdot a \]

12. Final simplification 7.5%

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

Reproduce

herbie shell --seed 2025084 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))