Quotient of sum of exps

Percentage Accurate: 98.8% → 99.0%

Time: 8.0s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.4183512390557513e+79
  2. b = 9.483160115154222e+144

Specification

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (fma (log (+ (exp a) (exp b))) -1.0 a)))

C

double code(double a, double b) {
	return exp(fma(log((exp(a) + exp(b))), -1.0, a));
}

Julia

function code(a, b)
	return exp(fma(log(Float64(exp(a) + exp(b))), -1.0, a))
end

Wolfram

code[a_, b_] := N[Exp[N[(N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -1.0 + a), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. inv-pow N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. prod-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-log.f64 98.8

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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.02:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.02) (/ 1.0 (+ 1.0 (exp (- a)))) (exp (- (log1p (exp b))))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.02) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = exp(-log1p(exp(b)));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.02) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = Math.exp(-Math.log1p(Math.exp(b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.02:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = math.exp(-math.log1p(math.exp(b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.02)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = exp(Float64(-log1p(exp(b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.02], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[(-N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0200000000000000004

   1. Initial program 96.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if 0.0200000000000000004 < (exp.f64 a)

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-exp.f64 99.1

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

   7. Applied rewrites 99.1%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{a} \cdot \frac{1}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (exp a) (/ 1.0 (+ (exp a) (exp b)))))

C

double code(double a, double b) {
	return exp(a) * (1.0 / (exp(a) + exp(b)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) * (1.0d0 / (exp(a) + exp(b)))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) * (1.0 / (Math.exp(a) + Math.exp(b)));
}

Python

def code(a, b):
	return math.exp(a) * (1.0 / (math.exp(a) + math.exp(b)))

Julia

function code(a, b)
	return Float64(exp(a) * Float64(1.0 / Float64(exp(a) + exp(b))))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) * (1.0 / (exp(a) + exp(b)));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] * N[(1.0 / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. div-inv N/A

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

   6. lift-+.f64 N/A

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

   7. flip3-+ N/A

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

   8. clear-num N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

   \[\leadsto e^{a} \cdot \frac{1}{e^{a} + e^{b}} \]
6. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 5: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.02:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.02) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.02) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.02d0) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.02) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.02:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.02)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.02)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.02], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0200000000000000004

   1. Initial program 96.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-exp.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if 0.0200000000000000004 < (exp.f64 a)

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.02:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.02:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.02) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.02) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.02d0) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.02) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.02:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.02)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.02)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.02], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0200000000000000004

   1. Initial program 96.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 93.9%

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

   if 0.0200000000000000004 < (exp.f64 a)

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.02:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{b} + 1} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ (exp b) 1.0)))

C

double code(double a, double b) {
	return 1.0 / (exp(b) + 1.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (exp(b) + 1.0d0)
end function

Java

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

Python

def code(a, b):
	return 1.0 / (math.exp(b) + 1.0)

Julia

function code(a, b)
	return Float64(1.0 / Float64(exp(b) + 1.0))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (exp(b) + 1.0);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 83.9

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

5. Applied rewrites 83.9%

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

6. Final simplification 83.9%

   \[\leadsto \frac{1}{e^{b} + 1} \]
7. Add Preprocessing

Alternative 8: 43.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (*
  (/ 1.0 (+ 1.0 (fma a (fma a (fma a 0.16666666666666666 0.5) 1.0) 1.0)))
  (fma a (fma a 0.5 1.0) 1.0)))

C

double code(double a, double b) {
	return (1.0 / (1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0))) * fma(a, fma(a, 0.5, 1.0), 1.0);
}

Julia

function code(a, b)
	return Float64(Float64(1.0 / Float64(1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0))) * fma(a, fma(a, 0.5, 1.0), 1.0))
end

Wolfram

code[a_, b_] := N[(N[(1.0 / N[(1.0 + N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * N[(a * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. div-inv N/A

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

   6. lift-+.f64 N/A

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

   7. flip3-+ N/A

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

   8. clear-num N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 61.6%

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

8. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 40.3

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

10. Applied rewrites 40.3%

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

11. Taylor expanded in a around 0

    \[\leadsto \frac{1}{\color{blue}{\left(1 + a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right)\right)} + 1} \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{1}{2}, 1\right), 1\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f64 43.1

       \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) + 1} \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right) \]

13. Applied rewrites 43.1%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)} + 1} \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right) \]

14. Final simplification 43.1%

    \[\leadsto \frac{1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right) \]
15. Add Preprocessing

Alternative 9: 39.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right)\\ t\_0 \cdot \frac{1}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma a (fma a 0.5 1.0) 1.0))) (* t_0 (/ 1.0 (+ 1.0 t_0)))))

C

double code(double a, double b) {
	double t_0 = fma(a, fma(a, 0.5, 1.0), 1.0);
	return t_0 * (1.0 / (1.0 + t_0));
}

Julia

function code(a, b)
	t_0 = fma(a, fma(a, 0.5, 1.0), 1.0)
	return Float64(t_0 * Float64(1.0 / Float64(1.0 + t_0)))
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * N[(a * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(t$95$0 * N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. div-inv N/A

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

   6. lift-+.f64 N/A

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

   7. flip3-+ N/A

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

   8. clear-num N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 61.6%

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

8. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 40.3

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

10. Applied rewrites 40.3%

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

11. Taylor expanded in a around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 41.2

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

13. Applied rewrites 41.2%

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

14. Final simplification 41.2%

    \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right) \cdot \frac{1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right)} \]
15. Add Preprocessing

Alternative 10: 39.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + 1} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 1.0)))

C

double code(double a, double b) {
	return 1.0 / (1.0 + 1.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + 1.0d0)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + 1.0);
}

Python

def code(a, b):
	return 1.0 / (1.0 + 1.0)

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + 1.0))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + 1.0);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Taylor expanded in b around 0

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

5. Applied rewrites 61.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 60.6%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 40.3%

    \[\leadsto \frac{\color{blue}{1}}{1 + 1} \]
12. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (exp (- b a)))))

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + exp((b - a)))
end function

Java

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

Python

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + exp(Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + exp((b - a)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024212 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ 1 (exp (- b a)))))

  (/ (exp a) (+ (exp a) (exp b))))