symmetry log of sum of exp

Percentage Accurate: 54.8% → 99.1%

Time: 8.7s

Alternatives: 17

Speedup: 1.4×

• could not determine a ground truth

  1. a = 2.601017187819476e+32
  2. b = 2.7445378585099158e+222

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: 54.8% 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: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -380:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{elif}\;a \leq -8.2 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -380.0)
   (/ b (+ 1.0 (exp a)))
   (if (<= a -8.2e-16) (log1p (exp a)) (log1p (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -380.0) {
		tmp = b / (1.0 + exp(a));
	} else if (a <= -8.2e-16) {
		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 (a <= -380.0) {
		tmp = b / (1.0 + Math.exp(a));
	} else if (a <= -8.2e-16) {
		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 a <= -380.0:
		tmp = b / (1.0 + math.exp(a))
	elif a <= -8.2e-16:
		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 (a <= -380.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	elseif (a <= -8.2e-16)
		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[a, -380.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -8.2e-16], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -380

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-exp.f64 98.6

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

   8. Applied rewrites 98.6%

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

   if -380 < a < -8.20000000000000012e-16

   1. Initial program 97.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 71.9

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

   5. Applied rewrites 71.9%

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

   if -8.20000000000000012e-16 < a

   1. Initial program 73.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lift-exp.f64 72.7

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

   5. Applied rewrites 72.7%

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

Alternative 2: 55.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{a} + e^{b}\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (log (+ (exp a) (exp b))) 5e-5)
   (* (+ (/ 0.5 b) 0.125) (* b b))
   (fma 0.5 b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (log((exp(a) + exp(b))) <= 5e-5) {
		tmp = ((0.5 / b) + 0.125) * (b * 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 (log(Float64(exp(a) + exp(b))) <= 5e-5)
		tmp = Float64(Float64(Float64(0.5 / b) + 0.125) * Float64(b * 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[N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e-5], N[(N[(N[(0.5 / b), $MachinePrecision] + 0.125), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(0.5 * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (exp.f64 a) (exp.f64 b))) < 5.00000000000000024e-5

   1. Initial program 9.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-+l+ N/A

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

      3. lower-log1p.f64 N/A

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

      4. lift-exp.f64 5.2

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

   8. Applied rewrites 5.2%

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

   9. Taylor expanded in b around 0

      \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right)} \]
   10. 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. lift-log.f64 2.9

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

   11. Applied rewrites 2.9%

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

   12. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 8.6

          \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right) \]

   14. Applied rewrites 8.6%

       \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot \color{blue}{b}\right) \]

   if 5.00000000000000024e-5 < (log.f64 (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. +-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 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in a around 0

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

      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. lower-log.f64 92.7

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

   8. Applied rewrites 92.7%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{a} + e^{b}\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% 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 56.9%

   \[\log \left(e^{a} + e^{b}\right) \]
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. +-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 78.3

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

5. Applied rewrites 78.3%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -57:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -57.0) (/ b (+ 1.0 (exp a))) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -57.0) {
		tmp = b / (1.0 + exp(a));
	} 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 <= (-57.0d0)) then
        tmp = b / (1.0d0 + exp(a))
    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 <= -57.0) {
		tmp = b / (1.0 + 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 <= -57.0:
		tmp = b / (1.0 + 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 <= -57.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	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 <= -57.0)
		tmp = b / (1.0 + exp(a));
	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, -57.0], N[(b / N[(1.0 + 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 < -57

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-exp.f64 98.6

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

   8. Applied rewrites 98.6%

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

   if -57 < a

   1. Initial program 74.7%

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

Alternative 5: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -57:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -57.0)
   (/ b (+ 1.0 (exp a)))
   (log (+ (exp a) (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -57.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((exp(a) + fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -57.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(a) + fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 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, -57.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -57

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-exp.f64 98.6

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

   8. Applied rewrites 98.6%

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

   if -57 < a

   1. Initial program 74.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto \log \left(e^{a} + \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log \left(e^{a} + \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + \color{blue}{1}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \log \left(e^{a} + \left(\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) \cdot b + 1\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), \color{blue}{b}, 1\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1, b, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot b\right) \cdot b + 1, b, 1\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot b, b, 1\right), b, 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot b + \frac{1}{2}, b, 1\right), b, 1\right)\right) \]

      8. lower-fma.f64 70.5

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right)\right) \]

   5. Applied rewrites 70.5%

      \[\leadsto \log \left(e^{a} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(1 + a\right) + \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05)
   (/ b (+ 1.0 (exp a)))
   (log (+ (+ 1.0 a) (fma (fma 0.5 b 1.0) b 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log(((1.0 + a) + fma(fma(0.5, b, 1.0), b, 1.0)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(Float64(1.0 + a) + fma(fma(0.5, b, 1.0), b, 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[N[Exp[a], $MachinePrecision], 0.05], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(1.0 + a), $MachinePrecision] + N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-exp.f64 98.6

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

   8. Applied rewrites 98.6%

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

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 74.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 70.4

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

   8. Applied rewrites 70.4%

      \[\leadsto \log \left(\color{blue}{\left(1 + a\right)} + \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -57:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -57.0)
   (/ b (+ 1.0 (exp a)))
   (log (+ (exp a) (fma (fma 0.5 b 1.0) b 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -57.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((exp(a) + fma(fma(0.5, b, 1.0), b, 1.0)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -57.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(a) + fma(fma(0.5, b, 1.0), b, 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, -57.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -57

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-exp.f64 98.6

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

   8. Applied rewrites 98.6%

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

   if -57 < a

   1. Initial program 74.7%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 71.2

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

   5. Applied rewrites 71.2%

      \[\leadsto \log \left(e^{a} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ 1.0 (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 (exp(a) <= 0.0) {
		tmp = b / (1.0 + 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 (exp(a) <= 0.0)
		tmp = Float64(b / Float64(1.0 + 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[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $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 (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lift-exp.f64 98.6

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

   8. Applied rewrites 98.6%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 74.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-+l+ N/A

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

      3. lower-log1p.f64 N/A

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

      4. lift-exp.f64 71.7

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in b around 0

      \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right)} \]
   10. 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. lift-log.f64 68.9

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

   11. Applied rewrites 68.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (* (+ (/ 0.5 b) 0.125) (* b 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 (exp(a) <= 0.0) {
		tmp = ((0.5 / b) + 0.125) * (b * 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 (exp(a) <= 0.0)
		tmp = Float64(Float64(Float64(0.5 / b) + 0.125) * Float64(b * 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[N[Exp[a], $MachinePrecision], 0.0], N[(N[(N[(0.5 / b), $MachinePrecision] + 0.125), $MachinePrecision] * N[(b * b), $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 (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-+l+ N/A

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

      3. lower-log1p.f64 N/A

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

      4. lift-exp.f64 3.5

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right)} \]
   10. 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. lift-log.f64 3.7

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

   11. Applied rewrites 3.7%

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

   12. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 13.4

          \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right) \]

   14. Applied rewrites 13.4%

       \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot \color{blue}{b}\right) \]

   if 0.0 < (exp.f64 a)

   1. Initial program 74.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-+l+ N/A

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

      3. lower-log1p.f64 N/A

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

      4. lift-exp.f64 71.7

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in b around 0

      \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right)} \]
   10. 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. lift-log.f64 68.9

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

   11. Applied rewrites 68.9%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), a, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (* (+ (/ 0.5 b) 0.125) (* b b))
   (fma (fma 0.125 a 0.5) a (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = ((0.5 / b) + 0.125) * (b * b);
	} else {
		tmp = fma(fma(0.125, a, 0.5), a, log(2.0));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[(N[(0.5 / b), $MachinePrecision] + 0.125), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * a + 0.5), $MachinePrecision] * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-+l+ N/A

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

      3. lower-log1p.f64 N/A

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

      4. lift-exp.f64 3.5

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right)} \]
   10. 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. lift-log.f64 3.7

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

   11. Applied rewrites 3.7%

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

   12. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 13.4

          \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right) \]

   14. Applied rewrites 13.4%

       \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot \color{blue}{b}\right) \]

   if 0.0 < (exp.f64 a)

   1. Initial program 74.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

   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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 69.9

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

   8. Applied rewrites 69.9%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), a, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (* (+ (/ 0.5 b) 0.125) (* b b)) (log1p (+ 1.0 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = ((0.5 / b) + 0.125) * (b * b);
	} else {
		tmp = log1p((1.0 + b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = ((0.5 / b) + 0.125) * (b * b);
	} else {
		tmp = Math.log1p((1.0 + b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = ((0.5 / b) + 0.125) * (b * b)
	else:
		tmp = math.log1p((1.0 + b))
	return tmp

Julia

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[(N[(0.5 / b), $MachinePrecision] + 0.125), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(1.0 + b), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-+l+ N/A

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

      3. lower-log1p.f64 N/A

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

      4. lift-exp.f64 3.5

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right)} \]
   10. 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. lift-log.f64 3.7

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

   11. Applied rewrites 3.7%

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

   12. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 13.4

          \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right) \]

   14. Applied rewrites 13.4%

       \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot \color{blue}{b}\right) \]

   if 0.0 < (exp.f64 a)

   1. Initial program 74.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 70.7

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-+l+ N/A

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

      3. lower-log1p.f64 N/A

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

      4. lift-exp.f64 71.7

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in b around 0

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

       1. lower-+.f64 67.5

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

   11. Applied rewrites 67.5%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05) (* (+ (/ 0.5 b) 0.125) (* b b)) (log1p (+ 1.0 a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = ((0.5 / b) + 0.125) * (b * b);
	} else {
		tmp = log1p((1.0 + a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.05) {
		tmp = ((0.5 / b) + 0.125) * (b * b);
	} else {
		tmp = Math.log1p((1.0 + a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.05:
		tmp = ((0.5 / b) + 0.125) * (b * b)
	else:
		tmp = math.log1p((1.0 + a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(Float64(Float64(0.5 / b) + 0.125) * Float64(b * 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[N[Exp[a], $MachinePrecision], 0.05], N[(N[(N[(0.5 / b), $MachinePrecision] + 0.125), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(1.0 + a), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-+l+ N/A

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

      3. lower-log1p.f64 N/A

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

      4. lift-exp.f64 3.5

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right)} \]
   10. 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. lift-log.f64 3.7

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

   11. Applied rewrites 3.7%

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

   12. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 13.4

          \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right) \]

   14. Applied rewrites 13.4%

       \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot \color{blue}{b}\right) \]

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 74.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 69.3

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

   8. Applied rewrites 69.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (* (+ (/ 0.5 b) 0.125) (* b b)) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = ((0.5 / b) + 0.125) * (b * b);
	} else {
		tmp = log(2.0);
	}
	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 (exp(a) <= 0.0d0) then
        tmp = ((0.5d0 / b) + 0.125d0) * (b * b)
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = ((0.5 / b) + 0.125) * (b * b);
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = ((0.5 / b) + 0.125) * (b * b)
	else:
		tmp = math.log(2.0)
	return tmp

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = ((0.5 / b) + 0.125) * (b * b);
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[(N[(0.5 / b), $MachinePrecision] + 0.125), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   3. Taylor expanded in b around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in a around 0

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

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

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

      2. associate-+l+ N/A

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

      3. lower-log1p.f64 N/A

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

      4. lift-exp.f64 3.5

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right)} \]
   10. 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. lift-log.f64 3.7

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

   11. Applied rewrites 3.7%

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

   12. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-+.f64 N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. lower-/.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 13.4

          \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right) \]

   14. Applied rewrites 13.4%

       \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot \color{blue}{b}\right) \]

   if 0.0 < (exp.f64 a)

   1. Initial program 74.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in a around 0

      \[\leadsto \log 2 \]
   7. Step-by-step derivation

      1. lower-log.f64 67.8

         \[\leadsto \log 2 \]

   8. Applied rewrites 67.8%

      \[\leadsto \log 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 9.0% accurate, 12.2× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return ((0.5 / b) + 0.125) * (b * 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) + 0.125d0) * (b * b)
end function

Java

assert a < b;
public static double code(double a, double b) {
	return ((0.5 / b) + 0.125) * (b * b);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return ((0.5 / b) + 0.125) * (b * b)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(Float64(0.5 / b) + 0.125) * Float64(b * b))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = ((0.5 / b) + 0.125) * (b * b);
end

Wolfram

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

Derivation

1. Initial program 56.9%

   \[\log \left(e^{a} + e^{b}\right) \]
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. +-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 78.3

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

5. Applied rewrites 78.3%

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

6. Taylor expanded in a around 0

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

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

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

   2. associate-+l+ N/A

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

   3. lower-log1p.f64 N/A

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

   4. lift-exp.f64 53.1

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

8. Applied rewrites 53.1%

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

9. Taylor expanded in b around 0

   \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right)} \]
10. 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. lift-log.f64 51.1

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

11. Applied rewrites 51.1%

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

12. Taylor expanded in b around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-+.f64 N/A

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

    5. associate-*r/ N/A

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

    6. metadata-eval N/A

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

    7. lower-/.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 6.2

       \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right) \]

14. Applied rewrites 6.2%

    \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot \color{blue}{b}\right) \]

15. Final simplification 6.2%

    \[\leadsto \left(\frac{0.5}{b} + 0.125\right) \cdot \left(b \cdot b\right) \]
16. Add Preprocessing

Alternative 15: 5.2% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

   \[\log \left(e^{a} + e^{b}\right) \]
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. +-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 78.3

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

5. Applied rewrites 78.3%

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

6. Taylor expanded in a around 0

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

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

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

   2. associate-+l+ N/A

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

   3. lower-log1p.f64 N/A

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

   4. lift-exp.f64 53.1

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

8. Applied rewrites 53.1%

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

9. Taylor expanded in b around 0

   \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot b\right)} \]
10. 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. lift-log.f64 51.1

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

11. Applied rewrites 51.1%

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

12. Taylor expanded in b around inf

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

    1. *-commutative N/A

       \[\leadsto {b}^{2} \cdot \frac{1}{8} \]

    2. lower-*.f64 N/A

       \[\leadsto {b}^{2} \cdot \frac{1}{8} \]

    3. pow2 N/A

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

    4. lift-*.f64 4.3

       \[\leadsto \left(b \cdot b\right) \cdot 0.125 \]

14. Applied rewrites 4.3%

    \[\leadsto \left(b \cdot b\right) \cdot 0.125 \]

15. Final simplification 4.3%

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

Alternative 16: 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 56.9%

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

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

   2. lift-exp.f64 52.4

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

5. Applied rewrites 52.4%

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

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-log.f64 51.3

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

8. Applied rewrites 51.3%

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

9. Taylor expanded in a around inf

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

    1. *-commutative N/A

       \[\leadsto {a}^{2} \cdot \frac{1}{8} \]

    2. lower-*.f64 N/A

       \[\leadsto {a}^{2} \cdot \frac{1}{8} \]

    3. unpow2 N/A

       \[\leadsto \left(a \cdot a\right) \cdot \frac{1}{8} \]

    4. lower-*.f64 4.0

       \[\leadsto \left(a \cdot a\right) \cdot 0.125 \]

11. Applied rewrites 4.0%

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

Alternative 17: 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 56.9%

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

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

   2. lift-exp.f64 52.4

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

5. Applied rewrites 52.4%

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

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 51.2

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

8. Applied rewrites 51.2%

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

9. Taylor expanded in a around inf

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

    1. lower-*.f64 6.4

       \[\leadsto 0.5 \cdot a \]

11. Applied rewrites 6.4%

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

Reproduce

herbie shell --seed 2025037 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))