Quotient of sum of exps

Percentage Accurate: 99.0% → 98.6%

Time: 8.2s

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: 99.0% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.8) (/ (exp a) (+ (exp a) 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.8) {
		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) <= 0.8d0) 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) <= 0.8) {
		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) <= 0.8:
		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) <= 0.8)
		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) <= 0.8)
		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], 0.8], 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) < 0.80000000000000004

   1. Initial program 97.2%

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

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

   if 0.80000000000000004 < (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 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% 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 3: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.8) (/ (exp a) 2.0) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.8) {
		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 (exp(a) <= 0.8d0) 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 (Math.exp(a) <= 0.8) {
		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 math.exp(a) <= 0.8:
		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 (exp(a) <= 0.8)
		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 (exp(a) <= 0.8)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.8], 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 (exp.f64 a) < 0.80000000000000004

   1. Initial program 97.2%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.8%

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

   if 0.80000000000000004 < (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 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(1 + b \cdot 0.5\right)\\ \mathbf{if}\;b \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 - t\_0}{4 - t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (+ 1.0 (* b 0.5)))))
   (if (<= b 4e+46)
     (/ (exp a) 2.0)
     (if (<= b 1.5e+154)
       (/ (- 2.0 t_0) (- 4.0 (* t_0 t_0)))
       (/ 2.0 (* b b))))))

C

double code(double a, double b) {
	double t_0 = b * (1.0 + (b * 0.5));
	double tmp;
	if (b <= 4e+46) {
		tmp = exp(a) / 2.0;
	} else if (b <= 1.5e+154) {
		tmp = (2.0 - t_0) / (4.0 - (t_0 * t_0));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = b * (1.0d0 + (b * 0.5d0))
    if (b <= 4d+46) then
        tmp = exp(a) / 2.0d0
    else if (b <= 1.5d+154) then
        tmp = (2.0d0 - t_0) / (4.0d0 - (t_0 * t_0))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (1.0 + (b * 0.5));
	double tmp;
	if (b <= 4e+46) {
		tmp = Math.exp(a) / 2.0;
	} else if (b <= 1.5e+154) {
		tmp = (2.0 - t_0) / (4.0 - (t_0 * t_0));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (1.0 + (b * 0.5))
	tmp = 0
	if b <= 4e+46:
		tmp = math.exp(a) / 2.0
	elif b <= 1.5e+154:
		tmp = (2.0 - t_0) / (4.0 - (t_0 * t_0))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(1.0 + Float64(b * 0.5)))
	tmp = 0.0
	if (b <= 4e+46)
		tmp = Float64(exp(a) / 2.0);
	elseif (b <= 1.5e+154)
		tmp = Float64(Float64(2.0 - t_0) / Float64(4.0 - Float64(t_0 * t_0)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (1.0 + (b * 0.5));
	tmp = 0.0;
	if (b <= 4e+46)
		tmp = exp(a) / 2.0;
	elseif (b <= 1.5e+154)
		tmp = (2.0 - t_0) / (4.0 - (t_0 * t_0));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4e+46], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[b, 1.5e+154], N[(N[(2.0 - t$95$0), $MachinePrecision] / N[(4.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4e46

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 78.6%

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

   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 77.1%

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

   if 4e46 < b < 1.50000000000000013e154

   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 6.4%

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

   8. Simplified 6.4%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f64 86.9%

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

   10. Applied egg-rr 86.9%

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

   if 1.50000000000000013e154 < 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 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 - b \cdot \left(1 + b \cdot 0.5\right)}{4 - \left(b \cdot \left(1 + b \cdot 0.5\right)\right) \cdot \left(b \cdot \left(1 + b \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(1 + b \cdot 0.5\right)\\ \mathbf{if}\;b \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 - t\_0}{4 - t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (+ 1.0 (* b 0.5)))))
   (if (<= b 4e+46)
     (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
     (if (<= b 1.5e+154)
       (/ (- 2.0 t_0) (- 4.0 (* t_0 t_0)))
       (/ 2.0 (* b b))))))

C

double code(double a, double b) {
	double t_0 = b * (1.0 + (b * 0.5));
	double tmp;
	if (b <= 4e+46) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 1.5e+154) {
		tmp = (2.0 - t_0) / (4.0 - (t_0 * t_0));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = b * (1.0d0 + (b * 0.5d0))
    if (b <= 4d+46) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else if (b <= 1.5d+154) then
        tmp = (2.0d0 - t_0) / (4.0d0 - (t_0 * t_0))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (1.0 + (b * 0.5));
	double tmp;
	if (b <= 4e+46) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 1.5e+154) {
		tmp = (2.0 - t_0) / (4.0 - (t_0 * t_0));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (1.0 + (b * 0.5))
	tmp = 0
	if b <= 4e+46:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	elif b <= 1.5e+154:
		tmp = (2.0 - t_0) / (4.0 - (t_0 * t_0))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(1.0 + Float64(b * 0.5)))
	tmp = 0.0
	if (b <= 4e+46)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	elseif (b <= 1.5e+154)
		tmp = Float64(Float64(2.0 - t_0) / Float64(4.0 - Float64(t_0 * t_0)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (1.0 + (b * 0.5));
	tmp = 0.0;
	if (b <= 4e+46)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	elseif (b <= 1.5e+154)
		tmp = (2.0 - t_0) / (4.0 - (t_0 * t_0));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 4e+46], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.5e+154], N[(N[(2.0 - t$95$0), $MachinePrecision] / N[(4.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4e46

   1. Initial program 98.5%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      7. exp-lowering-exp.f64 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 78.5%

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

   8. Taylor expanded in a around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 69.3%

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

   10. Simplified 69.3%

       \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}} \]

   if 4e46 < b < 1.50000000000000013e154

   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 6.4%

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

   8. Simplified 6.4%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      16. *-lowering-*.f64 86.9%

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

   10. Applied egg-rr 86.9%

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

   if 1.50000000000000013e154 < 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 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 - b \cdot \left(1 + b \cdot 0.5\right)}{4 - \left(b \cdot \left(1 + b \cdot 0.5\right)\right) \cdot \left(b \cdot \left(1 + b \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.8e+45)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (if (<= b 1.05e+103)
     (* a (* a (* a -0.020833333333333332)))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.8e+45) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 1.05e+103) {
		tmp = a * (a * (a * -0.020833333333333332));
	} 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 <= 2.8d+45) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else if (b <= 1.05d+103) then
        tmp = a * (a * (a * (-0.020833333333333332d0)))
    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 <= 2.8e+45) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 1.05e+103) {
		tmp = a * (a * (a * -0.020833333333333332));
	} 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 <= 2.8e+45:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	elif b <= 1.05e+103:
		tmp = a * (a * (a * -0.020833333333333332))
	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 <= 2.8e+45)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	elseif (b <= 1.05e+103)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	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 <= 2.8e+45)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	elseif (b <= 1.05e+103)
		tmp = a * (a * (a * -0.020833333333333332));
	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, 2.8e+45], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+103], N[(a * N[(a * N[(a * -0.020833333333333332), $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 < 2.7999999999999999e45

   1. Initial program 98.5%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      7. exp-lowering-exp.f64 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 78.5%

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

   8. Taylor expanded in a around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 69.3%

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

   10. Simplified 69.3%

       \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}} \]

   if 2.7999999999999999e45 < b < 1.0500000000000001e103

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 3.1%

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

   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. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 3.1%

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

   8. Simplified 3.1%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 52.9%

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

   11. Simplified 52.9%

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

   if 1.0500000000000001e103 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

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

      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 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 10^{+136}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.8e+45)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (if (<= b 1e+136) (* a (* a (* a -0.020833333333333332))) (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.8e+45) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 1e+136) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.8d+45) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else if (b <= 1d+136) then
        tmp = a * (a * (a * (-0.020833333333333332d0)))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.8e+45) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 1e+136) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.8e+45:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	elif b <= 1e+136:
		tmp = a * (a * (a * -0.020833333333333332))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.8e+45)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	elseif (b <= 1e+136)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.8e+45)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	elseif (b <= 1e+136)
		tmp = a * (a * (a * -0.020833333333333332));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.8e+45], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+136], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.7999999999999999e45

   1. Initial program 98.5%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      7. exp-lowering-exp.f64 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 78.5%

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

   8. Taylor expanded in a around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 69.3%

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

   10. Simplified 69.3%

       \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}} \]

   if 2.7999999999999999e45 < b < 1.00000000000000006e136

   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 19.3%

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

   6. Taylor expanded in a around 0

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.9%

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

   8. Simplified 2.9%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 41.6%

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

   11. Simplified 41.6%

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

   if 1.00000000000000006e136 < 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 90.9%

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

   8. Simplified 90.9%

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

   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 90.9%

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

   11. Simplified 90.9%

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

Alternative 8: 65.8% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{elif}\;b \leq 10^{+136}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.8e+45)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5)))))
   (if (<= b 1e+136) (* a (* a (* a -0.020833333333333332))) (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.8e+45) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else if (b <= 1e+136) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.8d+45) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
    else if (b <= 1d+136) then
        tmp = a * (a * (a * (-0.020833333333333332d0)))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.8e+45) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else if (b <= 1e+136) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.8e+45:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))
	elif b <= 1e+136:
		tmp = a * (a * (a * -0.020833333333333332))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.8e+45)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))));
	elseif (b <= 1e+136)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.8e+45)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	elseif (b <= 1e+136)
		tmp = a * (a * (a * -0.020833333333333332));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.8e+45], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+136], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.7999999999999999e45

   1. Initial program 98.5%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      7. exp-lowering-exp.f64 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 78.5%

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

   8. Taylor expanded in a around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 64.0%

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

   10. Simplified 64.0%

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

   if 2.7999999999999999e45 < b < 1.00000000000000006e136

   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 19.3%

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

   6. Taylor expanded in a around 0

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.9%

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

   8. Simplified 2.9%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 41.6%

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

   11. Simplified 41.6%

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

   if 1.00000000000000006e136 < 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 90.9%

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

   8. Simplified 90.9%

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

   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 90.9%

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

   11. Simplified 90.9%

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

Alternative 9: 58.1% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2400:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq 10^{+136}:\\ \;\;\;\;a \cdot \left(a \cdot \left(a \cdot -0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2400.0)
   (/ 1.0 (- 2.0 a))
   (if (<= b 1e+136) (* a (* a (* a -0.020833333333333332))) (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2400.0) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 1e+136) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2400.0d0) then
        tmp = 1.0d0 / (2.0d0 - a)
    else if (b <= 1d+136) then
        tmp = a * (a * (a * (-0.020833333333333332d0)))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2400.0) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 1e+136) {
		tmp = a * (a * (a * -0.020833333333333332));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2400.0:
		tmp = 1.0 / (2.0 - a)
	elif b <= 1e+136:
		tmp = a * (a * (a * -0.020833333333333332))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2400.0)
		tmp = Float64(1.0 / Float64(2.0 - a));
	elseif (b <= 1e+136)
		tmp = Float64(a * Float64(a * Float64(a * -0.020833333333333332)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2400.0)
		tmp = 1.0 / (2.0 - a);
	elseif (b <= 1e+136)
		tmp = a * (a * (a * -0.020833333333333332));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2400.0], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+136], N[(a * N[(a * N[(a * -0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2400

   1. Initial program 98.9%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      7. exp-lowering-exp.f64 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 79.9%

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

   8. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 55.3%

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

   10. Simplified 55.3%

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

   if 2400 < b < 1.00000000000000006e136

   1. Initial program 96.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 33.1%

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

   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. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.7%

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 36.7%

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

   11. Simplified 36.7%

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

   if 1.00000000000000006e136 < 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 90.9%

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

   8. Simplified 90.9%

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

   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 90.9%

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

   11. Simplified 90.9%

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

Alternative 10: 53.4% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5e+45) (/ 1.0 (- 2.0 a)) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e+45) {
		tmp = 1.0 / (2.0 - 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 <= 5d+45) then
        tmp = 1.0d0 / (2.0d0 - 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 <= 5e+45) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5e+45:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e+45)
		tmp = Float64(1.0 / Float64(2.0 - 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 <= 5e+45)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e+45], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5e45

   1. Initial program 98.5%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

      7. exp-lowering-exp.f64 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in b around 0

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

   7. Simplified 78.5%

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

   8. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 52.5%

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

   10. Simplified 52.5%

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

   if 5e45 < 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 64.5%

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

   8. Simplified 64.5%

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

   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 64.5%

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

   11. Simplified 64.5%

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

Alternative 11: 40.3% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -800:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -800.0) 0.5 (/ 1.0 (+ b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -800.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-800.0d0)) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / (b + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -800.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -800.0:
		tmp = 0.5
	else:
		tmp = 1.0 / (b + 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -800.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / Float64(b + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -800.0)
		tmp = 0.5;
	else
		tmp = 1.0 / (b + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -800.0], 0.5, N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -800

   1. Initial program 97.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 18.4%

      \[\leadsto \color{blue}{0.5} \]

   if -800 < b

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 78.7%

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

   5. Simplified 78.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 47.0%

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

   8. Simplified 47.0%

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

Alternative 12: 40.2% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (- 2.0 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 - a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 - a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 - a);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

   7. exp-lowering-exp.f64 98.8%

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in b around 0

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

7. Simplified 68.3%

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

8. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 41.6%

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

10. Simplified 41.6%

    \[\leadsto \frac{1}{\color{blue}{2 - a}} \]
11. Add Preprocessing

Alternative 13: 39.3% 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 81.8%

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

5. Simplified 81.8%

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

6. Taylor expanded in b around 0

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

8. Simplified 40.6%

   \[\leadsto \color{blue}{0.5} \]
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 2024163 
(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))))