Quotient of sum of exps

Percentage Accurate: 98.9% → 98.9%

Time: 10.0s

Alternatives: 16

Speedup: 2.7×

• could not determine a ground truth

  1. a = 7.311529345014676e+36
  2. b = -6.2970572017377586e-142

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: 98.9% 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: 98.9% 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 100.0%

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

Alternative 2: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999999998:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9999999998)
   (/ 1.0 (+ 1.0 (exp (- a))))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9999999998) {
		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) <= 0.9999999998d0) 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) <= 0.9999999998) {
		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) <= 0.9999999998:
		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) <= 0.9999999998)
		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) <= 0.9999999998)
		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], 0.9999999998], 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) < 0.9999999998

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. exp-lowering-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. neg-lowering-neg.f64 98.7

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

   7. Simplified 98.7%

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

   if 0.9999999998 < (exp.f64 a)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999999998:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = 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) <= 0.0d0) then
        tmp = 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) <= 0.0) {
		tmp = 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) <= 0.0:
		tmp = 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) <= 0.0)
		tmp = exp(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) <= 0.0)
		tmp = 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], 0.0], N[Exp[a], $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) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf

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

   7. Simplified 100.0%

      \[\leadsto e^{\color{blue}{a}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 98.7%

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

Alternative 4: 100.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. prod-exp N/A

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

   6. exp-lowering-exp.f64 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. log-lowering-log.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. exp-lowering-exp.f64 N/A

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

   11. exp-lowering-exp.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in a around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

   4. rem-exp-log N/A

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

   5. remove-double-div N/A

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

   6. exp-neg N/A

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

   7. associate-/r* N/A

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

   8. /-lowering-/.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. exp-neg N/A

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

   11. lft-mult-inverse N/A

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

   12. +-lowering-+.f64 N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. prod-exp N/A

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

   16. exp-lowering-exp.f64 N/A

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

   17. neg-mul-1 N/A

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

   18. unsub-neg N/A

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

   19. --lowering--.f64 99.9

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

7. Simplified 99.9%

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

   1. exp-diff N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. div-exp N/A

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

   5. exp-lowering-exp.f64 N/A

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

   6. --lowering--.f64 100.0

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

9. Applied egg-rr 100.0%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{e^{a - b}}}} \]
10. Add Preprocessing

Alternative 5: 50.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 5000:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 5000.0)
   (/ 1.0 (- 2.0 a))
   (* a (* a (* a -0.020833333333333332)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 5000.0) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = a * (a * (a * -0.020833333333333332));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(b) <= 5000.0d0) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = a * (a * (a * (-0.020833333333333332d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(b) <= 5000.0) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = a * (a * (a * -0.020833333333333332));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(b) <= 5000.0:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = a * (a * (a * -0.020833333333333332))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 5000.0)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(b) <= 5000.0)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = a * (a * (a * -0.020833333333333332));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[b], $MachinePrecision], 5000.0], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 5e3

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. exp-lowering-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. neg-lowering-neg.f64 79.4

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

   7. Simplified 79.4%

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

   8. Taylor expanded in a around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

         \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

      3. --lowering--.f64 54.8

         \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   10. Simplified 54.8%

       \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   if 5e3 < (exp.f64 b)

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 40.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.6

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, \color{blue}{a \cdot a}, 0.25\right), 0.5\right) \]

   8. Simplified 2.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(\color{blue}{{a}^{2}} \cdot a\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right) \cdot a} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       6. *-commutative N/A

          \[\leadsto a \cdot \color{blue}{\left({a}^{2} \cdot \frac{-1}{48}\right)} \]

       7. unpow2 N/A

          \[\leadsto a \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{-1}{48}\right) \]

       8. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       10. *-lowering-*.f64 49.3

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 49.3%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 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]

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. prod-exp N/A

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

   6. exp-lowering-exp.f64 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. log-lowering-log.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. exp-lowering-exp.f64 N/A

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

   11. exp-lowering-exp.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in a around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

   4. rem-exp-log N/A

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

   5. remove-double-div N/A

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

   6. exp-neg N/A

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

   7. associate-/r* N/A

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

   8. /-lowering-/.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. exp-neg N/A

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

   11. lft-mult-inverse N/A

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

   12. +-lowering-+.f64 N/A

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

   13. *-commutative N/A

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

   14. neg-mul-1 N/A

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

   15. prod-exp N/A

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

   16. exp-lowering-exp.f64 N/A

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

   17. neg-mul-1 N/A

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

   18. unsub-neg N/A

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

   19. --lowering--.f64 99.9

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

7. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
8. Add Preprocessing

Alternative 7: 84.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right)\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{-40}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{t\_0}{t\_0 \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma b 0.5 1.0) (* b (fma b (* b 0.5) b)) -4.0)))
   (if (<= b -2.4e-40)
     (exp a)
     (if (<= b 3.3e+51)
       (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
       (if (<= b 1.6e+77)
         (/ t_0 (* t_0 (fma b (fma b 0.5 1.0) 2.0)))
         (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0)))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(b, 0.5, 1.0), (b * fma(b, (b * 0.5), b)), -4.0);
	double tmp;
	if (b <= -2.4e-40) {
		tmp = exp(a);
	} else if (b <= 3.3e+51) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1.6e+77) {
		tmp = t_0 / (t_0 * fma(b, fma(b, 0.5, 1.0), 2.0));
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(b, 0.5, 1.0), Float64(b * fma(b, Float64(b * 0.5), b)), -4.0)
	tmp = 0.0
	if (b <= -2.4e-40)
		tmp = exp(a);
	elseif (b <= 3.3e+51)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1.6e+77)
		tmp = Float64(t_0 / Float64(t_0 * fma(b, fma(b, 0.5, 1.0), 2.0)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * 0.5 + 1.0), $MachinePrecision] * N[(b * N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision]}, If[LessEqual[b, -2.4e-40], N[Exp[a], $MachinePrecision], If[LessEqual[b, 3.3e+51], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+77], N[(t$95$0 / N[(t$95$0 * N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.39999999999999991e-40

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf

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

   7. Simplified 95.5%

      \[\leadsto e^{\color{blue}{a}} \]

   if -2.39999999999999991e-40 < b < 3.2999999999999997e51

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. exp-lowering-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. neg-lowering-neg.f64 94.0

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

   7. Simplified 94.0%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 85.0

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, -0.16666666666666666, 0.5\right)}, -1\right), 2\right)} \]

   10. Simplified 85.0%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}} \]

   if 3.2999999999999997e51 < b < 1.6000000000000001e77

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 4.3

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 2\right)} \]

   8. Simplified 4.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. flip-- N/A

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

      4. frac-times N/A

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

      5. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}} \]

   if 1.6000000000000001e77 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 93.4

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 2\right)} \]

   8. Simplified 93.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-40}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right)\\ \mathbf{if}\;b \leq 3.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{t\_0}{t\_0 \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma b 0.5 1.0) (* b (fma b (* b 0.5) b)) -4.0)))
   (if (<= b 3.2e+51)
     (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
     (if (<= b 1.6e+77)
       (/ t_0 (* t_0 (fma b (fma b 0.5 1.0) 2.0)))
       (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(b, 0.5, 1.0), (b * fma(b, (b * 0.5), b)), -4.0);
	double tmp;
	if (b <= 3.2e+51) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1.6e+77) {
		tmp = t_0 / (t_0 * fma(b, fma(b, 0.5, 1.0), 2.0));
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(b, 0.5, 1.0), Float64(b * fma(b, Float64(b * 0.5), b)), -4.0)
	tmp = 0.0
	if (b <= 3.2e+51)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1.6e+77)
		tmp = Float64(t_0 / Float64(t_0 * fma(b, fma(b, 0.5, 1.0), 2.0)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * 0.5 + 1.0), $MachinePrecision] * N[(b * N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision]}, If[LessEqual[b, 3.2e+51], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+77], N[(t$95$0 / N[(t$95$0 * N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.2000000000000002e51

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. exp-lowering-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. neg-lowering-neg.f64 77.7

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

   7. Simplified 77.7%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 69.6

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, -0.16666666666666666, 0.5\right)}, -1\right), 2\right)} \]

   10. Simplified 69.6%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}} \]

   if 3.2000000000000002e51 < b < 1.6000000000000001e77

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 4.3

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 2\right)} \]

   8. Simplified 4.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

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

      2. associate-/r/ N/A

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

      3. flip-- N/A

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

      4. frac-times N/A

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

      5. /-lowering-/.f64 N/A

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}} \]

   if 1.6000000000000001e77 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 93.4

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 2\right)} \]

   8. Simplified 93.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, b \cdot 0.5, b\right), -4\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8e+39)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (if (<= b 1.05e+103)
     (* a (* a (* a -0.020833333333333332)))
     (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8e+39) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1.05e+103) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8e+39)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1.05e+103)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8e+39], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+103], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7.99999999999999952e39

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. exp-lowering-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. neg-lowering-neg.f64 78.6

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

   7. Simplified 78.6%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 70.3

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, -0.16666666666666666, 0.5\right)}, -1\right), 2\right)} \]

   10. Simplified 70.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}} \]

   if 7.99999999999999952e39 < b < 1.0500000000000001e103

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 29.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.8

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, \color{blue}{a \cdot a}, 0.25\right), 0.5\right) \]

   8. Simplified 2.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(\color{blue}{{a}^{2}} \cdot a\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right) \cdot a} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       6. *-commutative N/A

          \[\leadsto a \cdot \color{blue}{\left({a}^{2} \cdot \frac{-1}{48}\right)} \]

       7. unpow2 N/A

          \[\leadsto a \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{-1}{48}\right) \]

       8. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       10. *-lowering-*.f64 54.5

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 54.5%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   if 1.0500000000000001e103 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 100.0

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 2\right)} \]

   8. Simplified 100.0%

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

Alternative 10: 68.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.65e+39)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (if (<= b 2.75e+153)
     (* a (* a (* a -0.020833333333333332)))
     (/ 1.0 (fma b (fma b 0.5 1.0) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65e+39) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 2.75e+153) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65e+39)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 2.75e+153)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65e+39], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.75e+153], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.6500000000000001e39

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. exp-lowering-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. neg-lowering-neg.f64 78.6

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

   7. Simplified 78.6%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 70.3

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, -0.16666666666666666, 0.5\right)}, -1\right), 2\right)} \]

   10. Simplified 70.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}} \]

   if 1.6500000000000001e39 < b < 2.7500000000000001e153

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 24.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.8

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, \color{blue}{a \cdot a}, 0.25\right), 0.5\right) \]

   8. Simplified 2.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(\color{blue}{{a}^{2}} \cdot a\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right) \cdot a} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       6. *-commutative N/A

          \[\leadsto a \cdot \color{blue}{\left({a}^{2} \cdot \frac{-1}{48}\right)} \]

       7. unpow2 N/A

          \[\leadsto a \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{-1}{48}\right) \]

       8. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       10. *-lowering-*.f64 60.7

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 60.7%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   if 2.7500000000000001e153 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 97.8

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 2\right)} \]

   8. Simplified 97.8%

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

Alternative 11: 66.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 940000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 940000.0)
   (/ 1.0 (fma a (fma a 0.5 -1.0) 2.0))
   (if (<= b 2.75e+153)
     (* a (* a (* a -0.020833333333333332)))
     (/ 1.0 (fma b (fma b 0.5 1.0) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 940000.0) {
		tmp = 1.0 / fma(a, fma(a, 0.5, -1.0), 2.0);
	} else if (b <= 2.75e+153) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 940000.0)
		tmp = Float64(1.0 / fma(a, fma(a, 0.5, -1.0), 2.0));
	elseif (b <= 2.75e+153)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 940000.0], N[(1.0 / N[(a * N[(a * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.75e+153], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 9.4e5

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. exp-lowering-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. neg-lowering-neg.f64 79.4

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

   7. Simplified 79.4%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 67.4

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0.5, -1\right)}, 2\right)} \]

   10. Simplified 67.4%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}} \]

   if 9.4e5 < b < 2.7500000000000001e153

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 31.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.7

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, \color{blue}{a \cdot a}, 0.25\right), 0.5\right) \]

   8. Simplified 2.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(\color{blue}{{a}^{2}} \cdot a\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right) \cdot a} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       6. *-commutative N/A

          \[\leadsto a \cdot \color{blue}{\left({a}^{2} \cdot \frac{-1}{48}\right)} \]

       7. unpow2 N/A

          \[\leadsto a \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{-1}{48}\right) \]

       8. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       10. *-lowering-*.f64 57.2

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 57.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   if 2.7500000000000001e153 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 97.8

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 2\right)} \]

   8. Simplified 97.8%

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

Alternative 12: 66.6% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 700:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 700.0)
   (/ 1.0 (fma a (fma a 0.5 -1.0) 2.0))
   (if (<= b 2.75e+153)
     (* a (* a (* a -0.020833333333333332)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 700.0) {
		tmp = 1.0 / fma(a, fma(a, 0.5, -1.0), 2.0);
	} else if (b <= 2.75e+153) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 700.0)
		tmp = Float64(1.0 / fma(a, fma(a, 0.5, -1.0), 2.0));
	elseif (b <= 2.75e+153)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 700.0], N[(1.0 / N[(a * N[(a * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.75e+153], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 700

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. exp-lowering-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. neg-lowering-neg.f64 79.4

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

   7. Simplified 79.4%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 67.4

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0.5, -1\right)}, 2\right)} \]

   10. Simplified 67.4%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}} \]

   if 700 < b < 2.7500000000000001e153

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 31.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.7

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, \color{blue}{a \cdot a}, 0.25\right), 0.5\right) \]

   8. Simplified 2.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(\color{blue}{{a}^{2}} \cdot a\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right) \cdot a} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       6. *-commutative N/A

          \[\leadsto a \cdot \color{blue}{\left({a}^{2} \cdot \frac{-1}{48}\right)} \]

       7. unpow2 N/A

          \[\leadsto a \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{-1}{48}\right) \]

       8. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       10. *-lowering-*.f64 57.2

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 57.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   if 2.7500000000000001e153 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 97.8

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 2\right)} \]

   8. Simplified 97.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 97.8

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

   11. Simplified 97.8%

       \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.2% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 550:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{+153}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 550.0)
   (/ 1.0 (- 2.0 a))
   (if (<= b 2.75e+153)
     (* a (* a (* a -0.020833333333333332)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 550.0) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 2.75e+153) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 550.0d0) then
        tmp = 1.0d0 / (2.0d0 - a)
    else if (b <= 2.75d+153) then
        tmp = a * (a * (a * (-0.020833333333333332d0)))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 550.0) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 2.75e+153) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 550.0:
		tmp = 1.0 / (2.0 - a)
	elif b <= 2.75e+153:
		tmp = a * (a * (a * -0.020833333333333332))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 550.0)
		tmp = Float64(1.0 / Float64(2.0 - a));
	elseif (b <= 2.75e+153)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 550.0)
		tmp = 1.0 / (2.0 - a);
	elseif (b <= 2.75e+153)
		tmp = a * (a * (a * -0.020833333333333332));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 550.0], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.75e+153], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 550

   1. Initial program 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. exp-lowering-exp.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. +-lowering-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. exp-lowering-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. neg-lowering-neg.f64 79.4

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

   7. Simplified 79.4%

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

   8. Taylor expanded in a around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

         \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

      3. --lowering--.f64 54.8

         \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   10. Simplified 54.8%

       \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   if 550 < b < 2.7500000000000001e153

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 31.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.7

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, \color{blue}{a \cdot a}, 0.25\right), 0.5\right) \]

   8. Simplified 2.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(\color{blue}{{a}^{2}} \cdot a\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right) \cdot a} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(\frac{-1}{48} \cdot {a}^{2}\right)} \]

       6. *-commutative N/A

          \[\leadsto a \cdot \color{blue}{\left({a}^{2} \cdot \frac{-1}{48}\right)} \]

       7. unpow2 N/A

          \[\leadsto a \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot \frac{-1}{48}\right) \]

       8. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(a \cdot \frac{-1}{48}\right)\right)} \]

       10. *-lowering-*.f64 57.2

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(a \cdot -0.020833333333333332\right)}\right) \]

   11. Simplified 57.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)} \]

   if 2.7500000000000001e153 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 97.8

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 2\right)} \]

   8. Simplified 97.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 97.8

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

   11. Simplified 97.8%

       \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 39.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (- 2.0 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 - a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 - a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 - a);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. prod-exp N/A

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

   6. exp-lowering-exp.f64 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. log-lowering-log.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. exp-lowering-exp.f64 N/A

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

   11. exp-lowering-exp.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

   4. remove-double-div N/A

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

   5. exp-neg N/A

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

   6. rem-exp-log N/A

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

   7. associate-/r* N/A

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

   8. /-lowering-/.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. exp-neg N/A

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

   12. lft-mult-inverse N/A

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

   13. *-rgt-identity N/A

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

   14. +-lowering-+.f64 N/A

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

   15. neg-mul-1 N/A

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

   16. exp-lowering-exp.f64 N/A

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

   17. neg-mul-1 N/A

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

   18. neg-lowering-neg.f64 68.2

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

7. Simplified 68.2%

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

8. Taylor expanded in a around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

      \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   3. --lowering--.f64 40.3

      \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

10. Simplified 40.3%

    \[\leadsto \frac{1}{\color{blue}{2 - a}} \]
11. Add Preprocessing

Alternative 15: 38.9% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, 0.25, 0.5\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma a 0.25 0.5))

C

double code(double a, double b) {
	return fma(a, 0.25, 0.5);
}

Julia

function code(a, b)
	return fma(a, 0.25, 0.5)
end

Wolfram

code[a_, b_] := N[(a * 0.25 + 0.5), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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. Simplified 68.2%

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

6. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot a} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

   3. accelerator-lowering-fma.f64 39.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 0.25, 0.5\right)} \]

8. Simplified 39.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, 0.25, 0.5\right)} \]
9. Add Preprocessing

Alternative 16: 38.8% accurate, 315.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5;
end

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in a around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-lowering-+.f64 N/A

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

   3. exp-lowering-exp.f64 81.3

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

5. Simplified 81.3%

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

6. Taylor expanded in b around 0

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

8. Simplified 39.0%

   \[\leadsto \color{blue}{0.5} \]
9. 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 2024198 
(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))))