symmetry log of sum of exp

Percentage Accurate: 54.0% → 98.7%

Time: 10.8s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. a = 2.0253972706599808e+222
  2. b = -1.1294392204939234e-204

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: 54.0% 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: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{b}{e^{a} - -1} + \mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(1, e^{-b}, e^{-a}\right)\right) - \log \left(e^{\left(-a\right) - b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e-16)
   (+ (/ b (- (exp a) -1.0)) (log1p (exp a)))
   (- (log (fma 1.0 (exp (- b)) (exp (- a)))) (log (exp (- (- a) b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e-16) {
		tmp = (b / (exp(a) - -1.0)) + log1p(exp(a));
	} else {
		tmp = log(fma(1.0, exp(-b), exp(-a))) - log(exp((-a - b)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e-16)
		tmp = Float64(Float64(b / Float64(exp(a) - -1.0)) + log1p(exp(a)));
	else
		tmp = Float64(log(fma(1.0, exp(Float64(-b)), exp(Float64(-a)))) - log(exp(Float64(Float64(-a) - 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[a, -2e-16], N[(N[(b / N[(N[Exp[a], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(1.0 * N[Exp[(-b)], $MachinePrecision] + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[Exp[N[((-a) - b), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e-16

   1. Initial program 8.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-exp.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -2e-16 < a

   1. Initial program 64.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-exp.f64 N/A

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

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

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

      5. flip-+ N/A

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

      6. sinh-cosh N/A

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

      7. sinh-cosh N/A

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

      8. sinh---cosh-rev N/A

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

      9. lift-exp.f64 N/A

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

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

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

      11. flip3-+ N/A

          \[\leadsto \log \left(\frac{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}{e^{\mathsf{neg}\left(a\right)}} + \color{blue}{\frac{{\cosh b}^{3} + {\sinh b}^{3}}{\cosh b \cdot \cosh b + \left(\sinh b \cdot \sinh b - \cosh b \cdot \sinh b\right)}}\right) \]

      12. flip3-+ N/A

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

      13. flip-+ N/A

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

      14. sinh---cosh-rev N/A

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

   4. Applied rewrites 62.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{b}{e^{a} - -1} + \mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(1, e^{-b}, e^{-a}\right)\right) - \log \left(e^{\left(-a\right) - b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{a} + e^{b}\right) \leq 0.6931471805599452:\\ \;\;\;\;0.5 \cdot b + \mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\sinh b + \cosh b\right)\\ \end{array} \end{array} \]

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))) 0.6931471805599452)
   (+ (* 0.5 b) (log1p (exp a)))
   (log1p (+ (sinh b) (cosh b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (log((exp(a) + exp(b))) <= 0.6931471805599452) {
		tmp = (0.5 * b) + log1p(exp(a));
	} else {
		tmp = log1p((sinh(b) + cosh(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))) <= 0.6931471805599452) {
		tmp = (0.5 * b) + Math.log1p(Math.exp(a));
	} else {
		tmp = Math.log1p((Math.sinh(b) + Math.cosh(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.log((math.exp(a) + math.exp(b))) <= 0.6931471805599452:
		tmp = (0.5 * b) + math.log1p(math.exp(a))
	else:
		tmp = math.log1p((math.sinh(b) + math.cosh(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (log(Float64(exp(a) + exp(b))) <= 0.6931471805599452)
		tmp = Float64(Float64(0.5 * b) + log1p(exp(a)));
	else
		tmp = log1p(Float64(sinh(b) + cosh(b)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.6931471805599452], N[(N[(0.5 * b), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(N[Sinh[b], $MachinePrecision] + N[Cosh[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (exp.f64 a) (exp.f64 b))) < 0.69314718055994518

   1. Initial program 10.2%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-exp.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 13.2%

      \[\leadsto 0.5 \cdot b + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   if 0.69314718055994518 < (log.f64 (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 93.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 91.3

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 91.2%

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

Alternative 3: 58.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{a} + e^{b}\right) \leq 0.6931471805599452:\\ \;\;\;\;0.5 \cdot b + \mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (log (+ (exp a) (exp b))) 0.6931471805599452)
   (+ (* 0.5 b) (log1p (exp a)))
   (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (log((exp(a) + exp(b))) <= 0.6931471805599452) {
		tmp = (0.5 * b) + log1p(exp(a));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.log((Math.exp(a) + Math.exp(b))) <= 0.6931471805599452) {
		tmp = (0.5 * b) + Math.log1p(Math.exp(a));
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.log((math.exp(a) + math.exp(b))) <= 0.6931471805599452:
		tmp = (0.5 * b) + math.log1p(math.exp(a))
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (log(Float64(exp(a) + exp(b))) <= 0.6931471805599452)
		tmp = Float64(Float64(0.5 * b) + log1p(exp(a)));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.6931471805599452], N[(N[(0.5 * b), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $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))) < 0.69314718055994518

   1. Initial program 10.2%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-exp.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 13.2%

      \[\leadsto 0.5 \cdot b + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   if 0.69314718055994518 < (log.f64 (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 93.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 91.3

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

   5. Applied rewrites 91.3%

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

Alternative 4: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{e^{a} - -1} + \mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1e-25)
   (+ (/ b (- (exp a) -1.0)) (log1p (exp a)))
   (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1e-25) {
		tmp = (b / (exp(a) - -1.0)) + log1p(exp(a));
	} else {
		tmp = log((exp(a) + exp(b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1e-25) {
		tmp = (b / (Math.exp(a) - -1.0)) + Math.log1p(Math.exp(a));
	} 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 <= -1e-25:
		tmp = (b / (math.exp(a) - -1.0)) + math.log1p(math.exp(a))
	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 <= -1e-25)
		tmp = Float64(Float64(b / Float64(exp(a) - -1.0)) + log1p(exp(a)));
	else
		tmp = log(Float64(exp(a) + 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[a, -1e-25], N[(N[(b / N[(N[Exp[a], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $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 < -1.00000000000000004e-25

   1. Initial program 10.4%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-exp.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -1.00000000000000004e-25 < a

   1. Initial program 65.1%

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

Alternative 5: 59.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot b + \mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 6.2e-9) (+ (* 0.5 b) (log1p (exp a))) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 6.2e-9) {
		tmp = (0.5 * b) + log1p(exp(a));
	} else {
		tmp = log((exp(a) + exp(b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 6.2e-9) {
		tmp = (0.5 * b) + Math.log1p(Math.exp(a));
	} 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 b <= 6.2e-9:
		tmp = (0.5 * b) + math.log1p(math.exp(a))
	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 (b <= 6.2e-9)
		tmp = Float64(Float64(0.5 * b) + log1p(exp(a)));
	else
		tmp = log(Float64(exp(a) + exp(b)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 6.2e-9], N[(N[(0.5 * b), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.2000000000000001e-9

   1. Initial program 47.3%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-log1p.f64 N/A

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

      17. lower-exp.f64 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 49.3%

      \[\leadsto 0.5 \cdot b + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   if 6.2000000000000001e-9 < b

   1. Initial program 70.9%

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

Alternative 6: 52.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.02e-155) (log1p (exp a)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.02e-155) {
		tmp = log1p(exp(a));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.02e-155) {
		tmp = Math.log1p(Math.exp(a));
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= 1.02e-155:
		tmp = math.log1p(math.exp(a))
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.02e-155)
		tmp = log1p(exp(a));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 1.02e-155], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0199999999999999e-155

   1. Initial program 44.7%

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

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

      2. lower-exp.f64 42.7

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

   5. Applied rewrites 42.7%

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

   if 1.0199999999999999e-155 < b

   1. Initial program 63.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 60.0

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

   5. Applied rewrites 60.0%

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

Alternative 7: 51.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.02e-155) (log1p (exp a)) (fma (fma 0.125 b 0.5) b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.02e-155) {
		tmp = log1p(exp(a));
	} else {
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= 1.02e-155)
		tmp = log1p(exp(a));
	else
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[b, 1.02e-155], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[(N[(0.125 * b + 0.5), $MachinePrecision] * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0199999999999999e-155

   1. Initial program 44.7%

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

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

      2. lower-exp.f64 42.7

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

   5. Applied rewrites 42.7%

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

   if 1.0199999999999999e-155 < b

   1. Initial program 63.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 60.0

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 55.9%

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

Alternative 8: 49.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in a around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 46.4

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

5. Applied rewrites 46.4%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 44.6%

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

Alternative 9: 49.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in b around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. metadata-eval N/A

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

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

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

   11. distribute-lft-neg-in N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-log1p.f64 N/A

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

   17. lower-exp.f64 71.6

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

5. Applied rewrites 71.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 44.3%

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

Alternative 10: 49.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in a around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 46.4

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

5. Applied rewrites 46.4%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 43.4%

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

Alternative 11: 48.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 44.4

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

5. Applied rewrites 44.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 43.1%

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

Alternative 12: 3.2% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
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 = (a * a) * 0.125d0
end function

Java

assert a < b;
public static double code(double a, double b) {
	return (a * a) * 0.125;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (a * a) * 0.125

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(a * a) * 0.125)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (a * a) * 0.125;
end

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 44.4

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

5. Applied rewrites 44.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 43.1%

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

9. Taylor expanded in a around inf

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

11. Applied rewrites 4.2%

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

Reproduce

herbie shell --seed 2025009 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))