Quotient of sum of exps

Percentage Accurate: 99.0% → 98.5%

Time: 6.0s

Alternatives: 5

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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-128) (/ (exp a) (+ (exp a) 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-128) {
		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) <= 5d-128) 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) <= 5e-128) {
		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) <= 5e-128:
		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) <= 5e-128)
		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) <= 5e-128)
		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], 5e-128], 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) < 5.0000000000000001e-128

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 100.0%

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

   if 5.0000000000000001e-128 < (exp.f64 a)

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
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. Final simplification 98.8%

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

Alternative 3: 53.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0033:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{{b}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.0033) (+ 0.5 (* a 0.25)) (/ -2.0 (pow b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.0033) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = -2.0 / pow(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 <= 0.0033d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = (-2.0d0) / (b ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 0.0033) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = -2.0 / Math.pow(b, 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.0033:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = -2.0 / math.pow(b, 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.0033)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(-2.0 / (b ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.0033)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = -2.0 / (b ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.0033], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0033

   1. Initial program 98.8%

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

   3. Taylor expanded in b around 0 79.9%

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

   4. Taylor expanded in a around 0 57.8%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 57.8%

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

   6. Simplified 57.8%

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

   if 0.0033 < b

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0 97.7%

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

      1. +-commutative 97.7%

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

      2. flip-+ 0.1%

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

      3. pow2 0.1%

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

      4. metadata-eval 0.1%

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

      5. expm1-udef 0.1%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - 1}{\color{blue}{\mathsf{expm1}\left(b\right)}}} \]

   5. Applied egg-rr 0.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(e^{b}\right)}^{2} - 1}{\mathsf{expm1}\left(b\right)}}} \]
   6. Step-by-step derivation

      1. unpow2 0.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b} \cdot e^{b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      2. prod-exp 0.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b + b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      3. expm1-def 0.1%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(b + b\right)}}{\mathsf{expm1}\left(b\right)}} \]

   7. Simplified 0.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}} \]

   8. Taylor expanded in b around 0 53.5%

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

   10. Simplified 53.5%

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

   11. Taylor expanded in b around inf 53.5%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0033:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{{b}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0 83.8%

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

4. Final simplification 83.8%

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

Alternative 5: 39.4% 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 83.8%

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

4. Taylor expanded in b around 0 39.5%

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

5. Final simplification 39.5%

   \[\leadsto 0.5 \]
6. Add Preprocessing

Developer target: 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 2024040 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

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