Quotient of sum of exps

Percentage Accurate: 98.9% → 98.6%

Time: 8.3s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = -2.00000000000000007e154

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: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - \mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99999)
   (/ (exp a) (+ (exp a) 1.0))
   (exp (- 0.0 (log1p (exp b))))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99999) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = exp((0.0 - log1p(exp(b))));
	}
	return tmp;
}

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.99999) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.exp((0.0 - Math.log1p(Math.exp(b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99999:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = math.exp((0.0 - math.log1p(math.exp(b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99999)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = exp(Float64(0.0 - log1p(exp(b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[N[(0.0 - N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999990000000000046

   1. Initial program 97.3%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 98.7%

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

   if 0.999990000000000046 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 99.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 99.6%

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

      1. inv-pow N/A

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

      2. pow-to-exp N/A

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

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

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(1 + e^{b}\right) \cdot -1\right)\right) \]

      4. rem-log-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\log \left(e^{\log \left(1 + e^{b}\right) \cdot -1}\right)\right) \]

      5. pow-to-exp N/A

         \[\leadsto \mathsf{exp.f64}\left(\log \left({\left(1 + e^{b}\right)}^{-1}\right)\right) \]

      6. inv-pow N/A

         \[\leadsto \mathsf{exp.f64}\left(\log \left(\frac{1}{1 + e^{b}}\right)\right) \]

      7. log-div N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log 1 - \log \left(1 + e^{b}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - \log \left(1 + e^{b}\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \log \left(1 + e^{b}\right)\right)\right) \]

      10. log1p-define N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(\mathsf{log1p}\left(e^{b}\right)\right)\right)\right) \]

      11. log1p-lowering-log1p.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log1p.f64}\left(\left(e^{b}\right)\right)\right)\right) \]

      12. exp-lowering-exp.f64 99.6%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(b\right)\right)\right)\right) \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{e^{0 - \mathsf{log1p}\left(e^{b}\right)}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\log \left(1 + e^{b}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\log \left(1 + e^{b}\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\left(\mathsf{log1p}\left(e^{b}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{log1p.f64}\left(\left(e^{b}\right)\right)\right)\right) \]

      5. exp-lowering-exp.f64 99.6%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(b\right)\right)\right)\right) \]

   9. Applied egg-rr 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;e^{0 - \mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99999) (/ (exp a) (+ (exp a) 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99999) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.99999d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.99999) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99999:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99999)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.99999)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999990000000000046

   1. Initial program 97.3%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 98.7%

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

   if 0.999990000000000046 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 99.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 98.8%

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

Alternative 4: 98.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -720000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -720000000.0) (/ (exp a) 2.0) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -720000000.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-720000000.0d0)) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -720000000.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -720000000.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -720000000.0)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -720000000.0)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -720000000.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.2e8

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{2}} \]

   if -7.2e8 < a

   1. Initial program 98.9%

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 98.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 98.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -720000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.5e+102)
   (/ (exp a) 2.0)
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.5e+102) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 9.5d+102) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.5e+102) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.5e+102:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.5e+102)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.5e+102)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.5e+102], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.4999999999999992e102

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 75.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 73.3%

      \[\leadsto \frac{e^{a}}{\color{blue}{2}} \]

   if 9.4999999999999992e102 < 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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 100.0%

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

Alternative 6: 63.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \frac{b \cdot b}{2} - 2\\ t_2 := \frac{2}{b \cdot b}\\ \mathbf{if}\;b \leq 7200:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+31}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\frac{8 + \frac{t\_0 \cdot t\_0}{8}}{4 + \frac{t\_1}{t\_2}}}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(0.25 \cdot \left(b \cdot b\right)\right) - 4}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (- (/ (* b b) 2.0) 2.0))
        (t_2 (/ 2.0 (* b b))))
   (if (<= b 7200.0)
     (+
      0.5
      (*
       a
       (+
        0.25
        (*
         (* a a)
         (+ -0.020833333333333332 (* (* a a) 0.0020833333333333333))))))
     (if (<= b 1.35e+31)
       (* -0.020833333333333332 (* a (* a a)))
       (if (<= b 1.6e+77)
         (/ 1.0 (/ (+ 8.0 (/ (* t_0 t_0) 8.0)) (+ 4.0 (/ t_1 t_2))))
         (if (<= b 1.35e+154)
           (/ 1.0 (/ (- (* (* b b) (* 0.25 (* b b))) 4.0) t_1))
           t_2))))))

C

double code(double a, double b) {
	double t_0 = b * (b * b);
	double t_1 = ((b * b) / 2.0) - 2.0;
	double t_2 = 2.0 / (b * b);
	double tmp;
	if (b <= 7200.0) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 1.35e+31) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else if (b <= 1.6e+77) {
		tmp = 1.0 / ((8.0 + ((t_0 * t_0) / 8.0)) / (4.0 + (t_1 / t_2)));
	} else if (b <= 1.35e+154) {
		tmp = 1.0 / ((((b * b) * (0.25 * (b * b))) - 4.0) / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = b * (b * b)
    t_1 = ((b * b) / 2.0d0) - 2.0d0
    t_2 = 2.0d0 / (b * b)
    if (b <= 7200.0d0) then
        tmp = 0.5d0 + (a * (0.25d0 + ((a * a) * ((-0.020833333333333332d0) + ((a * a) * 0.0020833333333333333d0)))))
    else if (b <= 1.35d+31) then
        tmp = (-0.020833333333333332d0) * (a * (a * a))
    else if (b <= 1.6d+77) then
        tmp = 1.0d0 / ((8.0d0 + ((t_0 * t_0) / 8.0d0)) / (4.0d0 + (t_1 / t_2)))
    else if (b <= 1.35d+154) then
        tmp = 1.0d0 / ((((b * b) * (0.25d0 * (b * b))) - 4.0d0) / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * b);
	double t_1 = ((b * b) / 2.0) - 2.0;
	double t_2 = 2.0 / (b * b);
	double tmp;
	if (b <= 7200.0) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 1.35e+31) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else if (b <= 1.6e+77) {
		tmp = 1.0 / ((8.0 + ((t_0 * t_0) / 8.0)) / (4.0 + (t_1 / t_2)));
	} else if (b <= 1.35e+154) {
		tmp = 1.0 / ((((b * b) * (0.25 * (b * b))) - 4.0) / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * b)
	t_1 = ((b * b) / 2.0) - 2.0
	t_2 = 2.0 / (b * b)
	tmp = 0
	if b <= 7200.0:
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))))
	elif b <= 1.35e+31:
		tmp = -0.020833333333333332 * (a * (a * a))
	elif b <= 1.6e+77:
		tmp = 1.0 / ((8.0 + ((t_0 * t_0) / 8.0)) / (4.0 + (t_1 / t_2)))
	elif b <= 1.35e+154:
		tmp = 1.0 / ((((b * b) * (0.25 * (b * b))) - 4.0) / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(Float64(b * b) / 2.0) - 2.0)
	t_2 = Float64(2.0 / Float64(b * b))
	tmp = 0.0
	if (b <= 7200.0)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(Float64(a * a) * Float64(-0.020833333333333332 + Float64(Float64(a * a) * 0.0020833333333333333))))));
	elseif (b <= 1.35e+31)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	elseif (b <= 1.6e+77)
		tmp = Float64(1.0 / Float64(Float64(8.0 + Float64(Float64(t_0 * t_0) / 8.0)) / Float64(4.0 + Float64(t_1 / t_2))));
	elseif (b <= 1.35e+154)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(b * b) * Float64(0.25 * Float64(b * b))) - 4.0) / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * b);
	t_1 = ((b * b) / 2.0) - 2.0;
	t_2 = 2.0 / (b * b);
	tmp = 0.0;
	if (b <= 7200.0)
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	elseif (b <= 1.35e+31)
		tmp = -0.020833333333333332 * (a * (a * a));
	elseif (b <= 1.6e+77)
		tmp = 1.0 / ((8.0 + ((t_0 * t_0) / 8.0)) / (4.0 + (t_1 / t_2)));
	elseif (b <= 1.35e+154)
		tmp = 1.0 / ((((b * b) * (0.25 * (b * b))) - 4.0) / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * b), $MachinePrecision] / 2.0), $MachinePrecision] - 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7200.0], N[(0.5 + N[(a * N[(0.25 + N[(N[(a * a), $MachinePrecision] * N[(-0.020833333333333332 + N[(N[(a * a), $MachinePrecision] * 0.0020833333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+31], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+77], N[(1.0 / N[(N[(8.0 + N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 8.0), $MachinePrecision]), $MachinePrecision] / N[(4.0 + N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(1.0 / N[(N[(N[(N[(b * b), $MachinePrecision] * N[(0.25 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < 7200

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 79.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{-1}{48} + \color{blue}{\frac{1}{480} \cdot {a}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \color{blue}{\left(\frac{1}{480} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({a}^{2} \cdot \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 52.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)} \]

   if 7200 < b < 1.34999999999999993e31

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 22.5%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 2.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\frac{-1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.9%

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

   9. Taylor expanded in a around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-lowering-*.f64 51.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   11. Simplified 51.5%

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

   if 1.34999999999999993e31 < 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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + \frac{1}{2} \cdot b\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot b\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 4.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 4.2%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(\frac{1}{2} \cdot {\color{blue}{b}}^{2}\right)\right)\right) \]

       2. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(\frac{1 \cdot {b}^{2}}{\color{blue}{2}}\right)\right)\right) \]

       3. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(\frac{{b}^{2}}{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left({b}^{2}\right), \color{blue}{2}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(b \cdot b\right), 2\right)\right)\right) \]

       6. *-lowering-*.f64 4.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), 2\right)\right)\right) \]

   11. Simplified 4.2%

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

       1. flip3-+ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{{2}^{3} + {\left(\frac{b \cdot b}{2}\right)}^{3}}{\color{blue}{2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({2}^{3} + {\left(\frac{b \cdot b}{2}\right)}^{3}\right), \color{blue}{\left(2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({2}^{3}\right), \left({\left(\frac{b \cdot b}{2}\right)}^{3}\right)\right), \left(\color{blue}{2 \cdot 2} + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \left({\left(\frac{b \cdot b}{2}\right)}^{3}\right)\right), \left(\color{blue}{2} \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

       5. cube-div N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \left(\frac{{\left(b \cdot b\right)}^{3}}{{2}^{3}}\right)\right), \left(2 \cdot \color{blue}{2} + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\left({\left(b \cdot b\right)}^{3}\right), \left({2}^{3}\right)\right)\right), \left(2 \cdot \color{blue}{2} + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

       7. unpow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\left({b}^{3} \cdot {b}^{3}\right), \left({2}^{3}\right)\right)\right), \left(2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({b}^{3}\right), \left({b}^{3}\right)\right), \left({2}^{3}\right)\right)\right), \left(2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(b \cdot \left(b \cdot b\right)\right), \left({b}^{3}\right)\right), \left({2}^{3}\right)\right)\right), \left(2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(b \cdot b\right)\right), \left({b}^{3}\right)\right), \left({2}^{3}\right)\right)\right), \left(2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \left({b}^{3}\right)\right), \left({2}^{3}\right)\right)\right), \left(2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \left(b \cdot \left(b \cdot b\right)\right)\right), \left({2}^{3}\right)\right)\right), \left(2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot b\right)\right)\right), \left({2}^{3}\right)\right)\right), \left(2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right), \left({2}^{3}\right)\right)\right), \left(2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right), 8\right)\right), \left(2 \cdot 2 + \left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right), 8\right)\right), \mathsf{+.f64}\left(\left(2 \cdot 2\right), \color{blue}{\left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot \frac{b \cdot b}{2}\right)}\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(8, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right), 8\right)\right), \mathsf{+.f64}\left(4, \left(\color{blue}{\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2}} - 2 \cdot \frac{b \cdot b}{2}\right)\right)\right)\right) \]

   13. Applied egg-rr 65.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{8 + \frac{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}{8}}{4 + \frac{\frac{b \cdot b}{2} - 2}{\frac{2}{b \cdot b}}}}} \]

   if 1.6000000000000001e77 < b < 1.35000000000000003e154

   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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + \frac{1}{2} \cdot b\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot b\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 6.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 6.7%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{2} \cdot {b}^{2}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(\frac{1}{2} \cdot {\color{blue}{b}}^{2}\right)\right)\right) \]

       2. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(\frac{1 \cdot {b}^{2}}{\color{blue}{2}}\right)\right)\right) \]

       3. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(\frac{{b}^{2}}{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left({b}^{2}\right), \color{blue}{2}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(b \cdot b\right), 2\right)\right)\right) \]

       6. *-lowering-*.f64 6.7%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), 2\right)\right)\right) \]

   11. Simplified 6.7%

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

       1. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{b \cdot b}{2} + \color{blue}{2}\right)\right) \]

       2. flip-+ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot 2}{\color{blue}{\frac{b \cdot b}{2} - 2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2} - 2 \cdot 2\right), \color{blue}{\left(\frac{b \cdot b}{2} - 2\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot b}{2} \cdot \frac{b \cdot b}{2}\right), \left(2 \cdot 2\right)\right), \left(\color{blue}{\frac{b \cdot b}{2}} - 2\right)\right)\right) \]

       5. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\left(\frac{b \cdot b}{2}\right)}^{2}\right), \left(2 \cdot 2\right)\right), \left(\frac{\color{blue}{b \cdot b}}{2} - 2\right)\right)\right) \]

       6. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\left(\left(b \cdot b\right) \cdot \frac{1}{2}\right)}^{2}\right), \left(2 \cdot 2\right)\right), \left(\frac{\color{blue}{b} \cdot b}{2} - 2\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\left(\left(b \cdot b\right) \cdot \frac{1}{2}\right)}^{2}\right), \left(2 \cdot 2\right)\right), \left(\frac{b \cdot b}{2} - 2\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({\left(b \cdot \left(b \cdot \frac{1}{2}\right)\right)}^{2}\right), \left(2 \cdot 2\right)\right), \left(\frac{\color{blue}{b} \cdot b}{2} - 2\right)\right)\right) \]

       9. unpow-prod-down N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left({b}^{2} \cdot {\left(b \cdot \frac{1}{2}\right)}^{2}\right), \left(2 \cdot 2\right)\right), \left(\frac{\color{blue}{b \cdot b}}{2} - 2\right)\right)\right) \]

       10. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(b \cdot b\right) \cdot {\left(b \cdot \frac{1}{2}\right)}^{2}\right), \left(2 \cdot 2\right)\right), \left(\frac{\color{blue}{b} \cdot b}{2} - 2\right)\right)\right) \]

       11. pow2 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\left(b \cdot b\right) \cdot \left(\left(b \cdot \frac{1}{2}\right) \cdot \left(b \cdot \frac{1}{2}\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\frac{b \cdot \color{blue}{b}}{2} - 2\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left(\left(b \cdot \frac{1}{2}\right) \cdot \left(b \cdot \frac{1}{2}\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\frac{\color{blue}{b \cdot b}}{2} - 2\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(b \cdot \frac{1}{2}\right) \cdot \left(b \cdot \frac{1}{2}\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\frac{\color{blue}{b} \cdot b}{2} - 2\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(\frac{1}{2} \cdot b\right) \cdot \left(b \cdot \frac{1}{2}\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\frac{b \cdot b}{2} - 2\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(\frac{1}{2} \cdot b\right) \cdot \left(\frac{1}{2} \cdot b\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\frac{b \cdot b}{2} - 2\right)\right)\right) \]

       16. swap-sqr N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\frac{b \cdot \color{blue}{b}}{2} - 2\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{4} \cdot \left(b \cdot b\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\frac{b \cdot b}{2} - 2\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{4}, \left(b \cdot b\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\frac{b \cdot \color{blue}{b}}{2} - 2\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(b, b\right)\right)\right), \left(2 \cdot 2\right)\right), \left(\frac{b \cdot b}{2} - 2\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(b, b\right)\right)\right), 4\right), \left(\frac{b \cdot b}{\color{blue}{2}} - 2\right)\right)\right) \]

       21. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(b, b\right)\right)\right), 4\right), \mathsf{\_.f64}\left(\left(\frac{b \cdot b}{2}\right), \color{blue}{2}\right)\right)\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(0.25 \cdot \left(b \cdot b\right)\right) - 4}{\frac{b \cdot b}{2} - 2}}} \]

   if 1.35000000000000003e154 < 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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + \frac{1}{2} \cdot b\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot b\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(b \cdot \color{blue}{b}\right)\right) \]

       3. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7200:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{+98}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7200.0)
   (+
    0.5
    (*
     a
     (+
      0.25
      (*
       (* a a)
       (+ -0.020833333333333332 (* (* a a) 0.0020833333333333333))))))
   (if (<= b 1.28e+98)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7200.0) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 1.28e+98) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 7200.0d0) then
        tmp = 0.5d0 + (a * (0.25d0 + ((a * a) * ((-0.020833333333333332d0) + ((a * a) * 0.0020833333333333333d0)))))
    else if (b <= 1.28d+98) then
        tmp = (-0.020833333333333332d0) * (a * (a * a))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7200.0) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 1.28e+98) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7200.0:
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))))
	elif b <= 1.28e+98:
		tmp = -0.020833333333333332 * (a * (a * a))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7200.0)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(Float64(a * a) * Float64(-0.020833333333333332 + Float64(Float64(a * a) * 0.0020833333333333333))))));
	elseif (b <= 1.28e+98)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7200.0)
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	elseif (b <= 1.28e+98)
		tmp = -0.020833333333333332 * (a * (a * a));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7200.0], N[(0.5 + N[(a * N[(0.25 + N[(N[(a * a), $MachinePrecision] * N[(-0.020833333333333332 + N[(N[(a * a), $MachinePrecision] * 0.0020833333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.28e+98], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7200

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 79.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{-1}{48} + \color{blue}{\frac{1}{480} \cdot {a}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \color{blue}{\left(\frac{1}{480} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({a}^{2} \cdot \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 52.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)} \]

   if 7200 < b < 1.28000000000000006e98

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 35.4%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 2.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\frac{-1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-lowering-*.f64 35.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   11. Simplified 35.4%

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

   if 1.28000000000000006e98 < 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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 96.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 96.1%

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

Alternative 8: 61.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7200:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{+98}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7200.0)
   (+ 0.5 (* a (+ 0.25 (* a (* a -0.020833333333333332)))))
   (if (<= b 1.28e+98)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7200.0) {
		tmp = 0.5 + (a * (0.25 + (a * (a * -0.020833333333333332))));
	} else if (b <= 1.28e+98) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 7200.0d0) then
        tmp = 0.5d0 + (a * (0.25d0 + (a * (a * (-0.020833333333333332d0)))))
    else if (b <= 1.28d+98) then
        tmp = (-0.020833333333333332d0) * (a * (a * a))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7200.0) {
		tmp = 0.5 + (a * (0.25 + (a * (a * -0.020833333333333332))));
	} else if (b <= 1.28e+98) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7200.0:
		tmp = 0.5 + (a * (0.25 + (a * (a * -0.020833333333333332))))
	elif b <= 1.28e+98:
		tmp = -0.020833333333333332 * (a * (a * a))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7200.0)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(a * Float64(a * -0.020833333333333332)))));
	elseif (b <= 1.28e+98)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7200.0)
		tmp = 0.5 + (a * (0.25 + (a * (a * -0.020833333333333332))));
	elseif (b <= 1.28e+98)
		tmp = -0.020833333333333332 * (a * (a * a));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7200.0], N[(0.5 + N[(a * N[(0.25 + N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.28e+98], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7200

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 79.8%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 52.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\frac{-1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 52.5%

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

   if 7200 < b < 1.28000000000000006e98

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 35.4%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 2.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\frac{-1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-lowering-*.f64 35.4%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   11. Simplified 35.4%

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

   if 1.28000000000000006e98 < 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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 96.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 96.1%

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

Alternative 9: 58.7% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7200:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7200.0)
   (+ 0.5 (* a (+ 0.25 (* a (* a -0.020833333333333332)))))
   (if (<= b 1.35e+154)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7200.0) {
		tmp = 0.5 + (a * (0.25 + (a * (a * -0.020833333333333332))));
	} else if (b <= 1.35e+154) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} 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 <= 7200.0d0) then
        tmp = 0.5d0 + (a * (0.25d0 + (a * (a * (-0.020833333333333332d0)))))
    else if (b <= 1.35d+154) then
        tmp = (-0.020833333333333332d0) * (a * (a * a))
    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 <= 7200.0) {
		tmp = 0.5 + (a * (0.25 + (a * (a * -0.020833333333333332))));
	} else if (b <= 1.35e+154) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7200.0:
		tmp = 0.5 + (a * (0.25 + (a * (a * -0.020833333333333332))))
	elif b <= 1.35e+154:
		tmp = -0.020833333333333332 * (a * (a * a))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7200.0)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(a * Float64(a * -0.020833333333333332)))));
	elseif (b <= 1.35e+154)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7200.0)
		tmp = 0.5 + (a * (0.25 + (a * (a * -0.020833333333333332))));
	elseif (b <= 1.35e+154)
		tmp = -0.020833333333333332 * (a * (a * a));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7200.0], N[(0.5 + N[(a * N[(0.25 + N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7200

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 79.8%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 52.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\frac{-1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 52.5%

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

   if 7200 < b < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 39.8%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 2.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\frac{-1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-lowering-*.f64 37.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   11. Simplified 37.0%

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

   if 1.35000000000000003e154 < 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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + \frac{1}{2} \cdot b\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot b\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(b \cdot \color{blue}{b}\right)\right) \]

       3. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.8% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7200:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7200.0)
   (+ 0.5 (* a 0.25))
   (if (<= b 1.35e+154)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7200.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.35e+154) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} 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 <= 7200.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 1.35d+154) then
        tmp = (-0.020833333333333332d0) * (a * (a * a))
    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 <= 7200.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.35e+154) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7200.0:
		tmp = 0.5 + (a * 0.25)
	elif b <= 1.35e+154:
		tmp = -0.020833333333333332 * (a * (a * a))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7200.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 1.35e+154)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7200.0)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 1.35e+154)
		tmp = -0.020833333333333332 * (a * (a * a));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7200.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 7200

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\color{blue}{a} \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \left(\frac{1}{1 + e^{b}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{\color{blue}{2}}}\right)\right)\right)\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{-1}{{\color{blue}{\left(1 + e^{b}\right)}}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({\left(1 + e^{b}\right)}^{2}\right)}\right)\right)\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\left(1 + e^{b}\right), \color{blue}{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right), 2\right)\right)\right)\right)\right) \]

      17. exp-lowering-exp.f64 73.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 73.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot a} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-lowering-*.f64 52.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{a}\right)\right) \]

   8. Simplified 52.4%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]

   if 7200 < b < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 39.8%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{-1}{48} \cdot {a}^{2}\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left({a}^{2} \cdot \color{blue}{\frac{-1}{48}}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(\left(a \cdot a\right) \cdot \frac{-1}{48}\right)\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \left(a \cdot \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \frac{-1}{48}\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 2.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\frac{-1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \color{blue}{\left({a}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \left(a \cdot {a}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \color{blue}{\left({a}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \left(a \cdot \color{blue}{a}\right)\right)\right) \]

       6. *-lowering-*.f64 37.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right) \]

   11. Simplified 37.0%

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

   if 1.35000000000000003e154 < 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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + \frac{1}{2} \cdot b\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot b\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(b \cdot \color{blue}{b}\right)\right) \]

       3. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7200:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.8% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.6) (+ 0.5 (* a 0.25)) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.6) {
		tmp = 0.5 + (a * 0.25);
	} 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 <= 1.6d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    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 <= 1.6) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.6:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.6)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.6)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.6], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\color{blue}{a} \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \left(\frac{1}{1 + e^{b}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{\color{blue}{2}}}\right)\right)\right)\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{-1}{{\color{blue}{\left(1 + e^{b}\right)}}^{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({\left(1 + e^{b}\right)}^{2}\right)}\right)\right)\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\left(1 + e^{b}\right), \color{blue}{2}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right), 2\right)\right)\right)\right)\right) \]

      17. exp-lowering-exp.f64 73.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right), 2\right)\right)\right)\right)\right) \]

   5. Simplified 73.2%

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

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot a} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-lowering-*.f64 53.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{a}\right)\right) \]

   8. Simplified 53.2%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]

   if 1.6000000000000001 < 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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 98.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 98.7%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + \frac{1}{2} \cdot b\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot b\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 47.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{b}\right)\right)\right)\right)\right) \]

   8. Simplified 47.9%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(2, \left(b \cdot \color{blue}{b}\right)\right) \]

       3. *-lowering-*.f64 47.9%

          \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right) \]

   11. Simplified 47.9%

       \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.3% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + a \cdot 0.25 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ 0.5 (* a 0.25)))

C

double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0 + (a * 0.25d0)
end function

Java

public static double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Python

def code(a, b):
	return 0.5 + (a * 0.25)

Julia

function code(a, b)
	return Float64(0.5 + Float64(a * 0.25))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 + (a * 0.25);
end

Wolfram

code[a_, b_] := N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\color{blue}{a} \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(a \cdot \left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{1 + e^{b}} - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \left(\frac{1}{1 + e^{b}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{1}{1 + e^{b}}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{2}}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{b}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{{\left(1 + e^{b}\right)}^{\color{blue}{2}}}\right)\right)\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{{\left(1 + e^{b}\right)}^{2}}}\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \left(\frac{-1}{{\color{blue}{\left(1 + e^{b}\right)}}^{2}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \color{blue}{\left({\left(1 + e^{b}\right)}^{2}\right)}\right)\right)\right)\right) \]

   15. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\left(1 + e^{b}\right), \color{blue}{2}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right), 2\right)\right)\right)\right)\right) \]

   17. exp-lowering-exp.f64 80.4%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right), \mathsf{/.f64}\left(-1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right), 2\right)\right)\right)\right)\right) \]

5. Simplified 80.4%

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

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot a} \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

   2. *-lowering-*.f64 38.9%

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{a}\right)\right) \]

8. Simplified 38.9%

   \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]

9. Final simplification 38.9%

   \[\leadsto 0.5 + a \cdot 0.25 \]
10. Add Preprocessing

Alternative 13: 39.2% accurate, 305.0× 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 98.8%

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

3. Taylor expanded in a around 0

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

   3. exp-lowering-exp.f64 80.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

5. Simplified 80.0%

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

6. Taylor expanded in b around 0

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

8. Simplified 38.6%

   \[\leadsto \color{blue}{0.5} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.9× 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 2024158 
(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))))