Quotient of sum of exps

Percentage Accurate: 98.9% → 98.0%

Time: 7.5s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -400000000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -400000000000.0) (/ (exp a) 2.0) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -400000000000.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-400000000000.0d0)) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -400000000000.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -400000000000.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -400000000000.0)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -400000000000.0)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -400000000000.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4e11

   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 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 100.0%

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

   if -4e11 < a

   1. Initial program 97.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 99.2%

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

   5. Simplified 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -400000000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

Alternative 3: 77.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{+96}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.9e+96) (/ (exp a) 2.0) (/ -4.0 (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.9e+96) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.9d+96) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (-4.0d0) / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.9e+96) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.9e+96:
		tmp = math.exp(a) / 2.0
	else:
		tmp = -4.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.9e+96)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(-4.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.9e+96)
		tmp = exp(a) / 2.0;
	else
		tmp = -4.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.9e+96], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(-4.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9000000000000001e96

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 71.3%

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

   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 70.6%

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

   if 1.9000000000000001e96 < b

   1. Initial program 96.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. *-commutative N/A

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

      5. *-lowering-*.f64 74.3%

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

   8. Simplified 74.3%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2 - 4 \cdot \frac{1}{b}}{{b}^{2}}} \]
   10. Step-by-step derivation

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

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

       2. sub-neg N/A

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

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

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

       4. associate-*r/ N/A

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{4}{b}\right)\right)\right), \left({b}^{2}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(4\right)}{b}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), b\right)\right), \left({b}^{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left({b}^{2}\right)\right) \]

       9. unpow2 N/A

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

       10. *-lowering-*.f64 74.3%

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

   11. Simplified 74.3%

       \[\leadsto \color{blue}{\frac{2 + \frac{-4}{b}}{b \cdot b}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{\frac{-4}{{b}^{3}}} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-4, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 96.7%

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

   14. Simplified 96.7%

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

Alternative 4: 59.2% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(1 + b \cdot 0.5\right)\\ \mathbf{if}\;b \leq -0.05:\\ \;\;\;\;\frac{1}{2 + \frac{b}{1 + b \cdot \left(-0.5 + b \cdot 0.25\right)}}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{4 + t\_0 \cdot \left(t\_0 + -2\right)}{8 + t\_0 \cdot \left(t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (+ 1.0 (* b 0.5)))))
   (if (<= b -0.05)
     (/ 1.0 (+ 2.0 (/ b (+ 1.0 (* b (+ -0.5 (* b 0.25)))))))
     (if (<= b 1.6e+77)
       (/ (+ 4.0 (* t_0 (+ t_0 -2.0))) (+ 8.0 (* t_0 (* t_0 t_0))))
       (/ -4.0 (* b (* b b)))))))

C

double code(double a, double b) {
	double t_0 = b * (1.0 + (b * 0.5));
	double tmp;
	if (b <= -0.05) {
		tmp = 1.0 / (2.0 + (b / (1.0 + (b * (-0.5 + (b * 0.25))))));
	} else if (b <= 1.6e+77) {
		tmp = (4.0 + (t_0 * (t_0 + -2.0))) / (8.0 + (t_0 * (t_0 * t_0)));
	} else {
		tmp = -4.0 / (b * (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 <= (-0.05d0)) then
        tmp = 1.0d0 / (2.0d0 + (b / (1.0d0 + (b * ((-0.5d0) + (b * 0.25d0))))))
    else if (b <= 1.6d+77) then
        tmp = (4.0d0 + (t_0 * (t_0 + (-2.0d0)))) / (8.0d0 + (t_0 * (t_0 * t_0)))
    else
        tmp = (-4.0d0) / (b * (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 <= -0.05) {
		tmp = 1.0 / (2.0 + (b / (1.0 + (b * (-0.5 + (b * 0.25))))));
	} else if (b <= 1.6e+77) {
		tmp = (4.0 + (t_0 * (t_0 + -2.0))) / (8.0 + (t_0 * (t_0 * t_0)));
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (1.0 + (b * 0.5))
	tmp = 0
	if b <= -0.05:
		tmp = 1.0 / (2.0 + (b / (1.0 + (b * (-0.5 + (b * 0.25))))))
	elif b <= 1.6e+77:
		tmp = (4.0 + (t_0 * (t_0 + -2.0))) / (8.0 + (t_0 * (t_0 * t_0)))
	else:
		tmp = -4.0 / (b * (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 <= -0.05)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b / Float64(1.0 + Float64(b * Float64(-0.5 + Float64(b * 0.25)))))));
	elseif (b <= 1.6e+77)
		tmp = Float64(Float64(4.0 + Float64(t_0 * Float64(t_0 + -2.0))) / Float64(8.0 + Float64(t_0 * Float64(t_0 * t_0))));
	else
		tmp = Float64(-4.0 / Float64(b * 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 <= -0.05)
		tmp = 1.0 / (2.0 + (b / (1.0 + (b * (-0.5 + (b * 0.25))))));
	elseif (b <= 1.6e+77)
		tmp = (4.0 + (t_0 * (t_0 + -2.0))) / (8.0 + (t_0 * (t_0 * t_0)));
	else
		tmp = -4.0 / (b * (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, -0.05], N[(1.0 / N[(2.0 + N[(b / N[(1.0 + N[(b * N[(-0.5 + N[(b * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e+77], N[(N[(4.0 + N[(t$95$0 * N[(t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(8.0 + N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.050000000000000003

   1. Initial program 96.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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. *-commutative N/A

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

      5. *-lowering-*.f64 5.4%

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

   8. Simplified 5.4%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 5.4%

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

   10. Applied egg-rr 5.4%

       \[\leadsto \frac{1}{2 + \color{blue}{\frac{b}{\frac{1}{1 + 0.5 \cdot b}}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{4} \cdot b - \frac{1}{2}\right)\right)}\right)\right)\right) \]
   12. Step-by-step derivation

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

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

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

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

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

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

       7. *-lowering-*.f64 19.2%

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

   13. Simplified 19.2%

       \[\leadsto \frac{1}{2 + \frac{b}{\color{blue}{1 + b \cdot \left(-0.5 + 0.25 \cdot b\right)}}} \]

   if -0.050000000000000003 < b < 1.6000000000000001e77

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 68.0%

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

   5. Simplified 68.0%

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

   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. *-commutative N/A

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

      5. *-lowering-*.f64 55.7%

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

   8. Simplified 55.7%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   10. Applied egg-rr 61.0%

       \[\leadsto \color{blue}{\frac{4 + \left(b \cdot \left(1 + 0.5 \cdot b\right)\right) \cdot \left(b \cdot \left(1 + 0.5 \cdot b\right) + -2\right)}{8 + \left(b \cdot \left(1 + 0.5 \cdot b\right)\right) \cdot \left(\left(b \cdot \left(1 + 0.5 \cdot b\right)\right) \cdot \left(b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)}} \]

   if 1.6000000000000001e77 < b

   1. Initial program 96.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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. *-commutative N/A

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

      5. *-lowering-*.f64 71.8%

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

   8. Simplified 71.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2 - 4 \cdot \frac{1}{b}}{{b}^{2}}} \]
   10. Step-by-step derivation

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

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

       2. sub-neg N/A

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

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

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

       4. associate-*r/ N/A

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{4}{b}\right)\right)\right), \left({b}^{2}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(4\right)}{b}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), b\right)\right), \left({b}^{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left({b}^{2}\right)\right) \]

       9. unpow2 N/A

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

       10. *-lowering-*.f64 71.8%

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

   11. Simplified 71.8%

       \[\leadsto \color{blue}{\frac{2 + \frac{-4}{b}}{b \cdot b}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{\frac{-4}{{b}^{3}}} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-4, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 93.5%

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

   14. Simplified 93.5%

       \[\leadsto \color{blue}{\frac{-4}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.05:\\ \;\;\;\;\frac{1}{2 + \frac{b}{1 + b \cdot \left(-0.5 + b \cdot 0.25\right)}}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{4 + \left(b \cdot \left(1 + b \cdot 0.5\right)\right) \cdot \left(b \cdot \left(1 + b \cdot 0.5\right) + -2\right)}{8 + \left(b \cdot \left(1 + b \cdot 0.5\right)\right) \cdot \left(\left(b \cdot \left(1 + b \cdot 0.5\right)\right) \cdot \left(b \cdot \left(1 + b \cdot 0.5\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;\frac{1}{2 + \frac{b}{1 + b \cdot \left(-0.5 + b \cdot 0.25\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.25)
   (/ 1.0 (+ 2.0 (/ b (+ 1.0 (* b (+ -0.5 (* b 0.25)))))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.25) {
		tmp = 1.0 / (2.0 + (b / (1.0 + (b * (-0.5 + (b * 0.25))))));
	} 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 <= (-1.25d0)) then
        tmp = 1.0d0 / (2.0d0 + (b / (1.0d0 + (b * ((-0.5d0) + (b * 0.25d0))))))
    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 <= -1.25) {
		tmp = 1.0 / (2.0 + (b / (1.0 + (b * (-0.5 + (b * 0.25))))));
	} 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 <= -1.25:
		tmp = 1.0 / (2.0 + (b / (1.0 + (b * (-0.5 + (b * 0.25))))))
	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 <= -1.25)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b / Float64(1.0 + Float64(b * Float64(-0.5 + Float64(b * 0.25)))))));
	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 <= -1.25)
		tmp = 1.0 / (2.0 + (b / (1.0 + (b * (-0.5 + (b * 0.25))))));
	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, -1.25], N[(1.0 / N[(2.0 + N[(b / N[(1.0 + N[(b * N[(-0.5 + N[(b * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 2 regimes

2. if b < -1.25

   1. Initial program 96.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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. *-commutative N/A

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

      5. *-lowering-*.f64 4.7%

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

   8. Simplified 4.7%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 4.7%

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

   10. Applied egg-rr 4.7%

       \[\leadsto \frac{1}{2 + \color{blue}{\frac{b}{\frac{1}{1 + 0.5 \cdot b}}}} \]

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{4} \cdot b - \frac{1}{2}\right)\right)}\right)\right)\right) \]
   12. Step-by-step derivation

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

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

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

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

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

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

       7. *-lowering-*.f64 18.8%

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

   13. Simplified 18.8%

       \[\leadsto \frac{1}{2 + \frac{b}{\color{blue}{1 + b \cdot \left(-0.5 + 0.25 \cdot b\right)}}} \]

   if -1.25 < 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 77.1%

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

   5. Simplified 77.1%

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

   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 66.3%

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

   8. Simplified 66.3%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.25:\\ \;\;\;\;\frac{1}{2 + \frac{b}{1 + b \cdot \left(-0.5 + b \cdot 0.25\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2.0)
   0.5
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		tmp = 0.5;
	} 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 <= (-2.0d0)) then
        tmp = 0.5d0
    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 <= -2.0) {
		tmp = 0.5;
	} 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 <= -2.0:
		tmp = 0.5
	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 <= -2.0)
		tmp = 0.5;
	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 <= -2.0)
		tmp = 0.5;
	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, -2.0], 0.5, N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2

   1. Initial program 96.4%

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

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

   8. Simplified 18.8%

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

   if -2 < 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 77.1%

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

   5. Simplified 77.1%

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

   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 66.3%

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

   8. Simplified 66.3%

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

Alternative 7: 56.6% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2450:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2450.0) 0.5 (/ -4.0 (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2450.0) {
		tmp = 0.5;
	} else {
		tmp = -4.0 / (b * (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 <= 2450.0d0) then
        tmp = 0.5d0
    else
        tmp = (-4.0d0) / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2450.0) {
		tmp = 0.5;
	} else {
		tmp = -4.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2450.0:
		tmp = 0.5
	else:
		tmp = -4.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2450.0)
		tmp = 0.5;
	else
		tmp = Float64(-4.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2450.0)
		tmp = 0.5;
	else
		tmp = -4.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2450.0], 0.5, N[(-4.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2450

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 74.7%

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

   5. Simplified 74.7%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 48.9%

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

   if 2450 < b

   1. Initial program 97.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. *-commutative N/A

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

      5. *-lowering-*.f64 55.3%

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

   8. Simplified 55.3%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{2 - 4 \cdot \frac{1}{b}}{{b}^{2}}} \]
   10. Step-by-step derivation

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

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

       2. sub-neg N/A

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

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

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

       4. associate-*r/ N/A

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

       5. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{4}{b}\right)\right)\right), \left({b}^{2}\right)\right) \]

       6. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(4\right)}{b}\right)\right), \left({b}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(4\right)\right), b\right)\right), \left({b}^{2}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(-4, b\right)\right), \left({b}^{2}\right)\right) \]

       9. unpow2 N/A

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

       10. *-lowering-*.f64 55.3%

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

   11. Simplified 55.3%

       \[\leadsto \color{blue}{\frac{2 + \frac{-4}{b}}{b \cdot b}} \]

   12. Taylor expanded in b around 0

       \[\leadsto \color{blue}{\frac{-4}{{b}^{3}}} \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(-4, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 71.9%

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

   14. Simplified 71.9%

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

Alternative 8: 52.0% accurate, 30.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. exp-lowering-exp.f64 74.7%

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

   5. Simplified 74.7%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 48.9%

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

   if 2 < b

   1. Initial program 97.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. *-commutative N/A

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

      5. *-lowering-*.f64 55.3%

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

   8. Simplified 55.3%

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

   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 55.3%

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

   11. Simplified 55.3%

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

Alternative 9: 38.2% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.4%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 82.0%

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

5. Simplified 82.0%

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

6. Taylor expanded in b around 0

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

8. Simplified 35.7%

   \[\leadsto \color{blue}{0.5} \]
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 2024155 
(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))))