Quotient of sum of exps

Percentage Accurate: 99.0% → 100.0%

Time: 7.2s

Alternatives: 5

Speedup: 2.9×

• 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: 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]

Derivation

1. Initial program 99.6%

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

   1. add-exp-log 99.6%

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

   2. div-exp 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in a around -inf 99.6%

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

   1. exp-neg 99.6%

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

   2. neg-mul-1 99.6%

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

   3. prod-exp 99.6%

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

   4. rem-exp-log 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-in 72.2%

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

   7. rec-exp 72.2%

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

   8. rgt-mult-inverse 100.0%

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

   9. prod-exp 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}}} \]

      2. pow1/3 100.0%

         \[\leadsto \color{blue}{{\left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 100.0%

         \[\leadsto \color{blue}{e^{\log \left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot 0.3333333333333333}} \]

      4. pow3 100.0%

         \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 100.0%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \cdot 0.3333333333333333} \]

      6. log-div 100.0%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{a}\right) - \log \left(e^{a} + e^{b}\right)\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 100.0%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{a} - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 0.3333333333333333} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 0.3333333333333333}} \]

   5. Taylor expanded in a around inf 100.0%

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

   if 0.0 < (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 99.2%

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

4. Final simplification 99.4%

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

Alternative 3: 51.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 1:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1.0) (exp a) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1.0) {
		tmp = exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	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) <= 1.0d0) then
        tmp = exp(a)
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 1.0:
		tmp = math.exp(a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1.0)
		tmp = exp(a);
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1.0)
		tmp = exp(a);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1.0], N[Exp[a], $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1

   1. Initial program 100.0%

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

      1. add-cbrt-cube 99.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}}} \]

      2. pow1/3 99.9%

         \[\leadsto \color{blue}{{\left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 99.5%

         \[\leadsto \color{blue}{e^{\log \left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot 0.3333333333333333}} \]

      4. pow3 99.5%

         \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 99.5%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \cdot 0.3333333333333333} \]

      6. log-div 99.5%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(\log \left(e^{a}\right) - \log \left(e^{a} + e^{b}\right)\right)}\right) \cdot 0.3333333333333333} \]

      7. add-log-exp 99.5%

         \[\leadsto e^{\left(3 \cdot \left(\color{blue}{a} - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 0.3333333333333333} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 0.3333333333333333}} \]

   5. Taylor expanded in a around inf 53.0%

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

   if 1 < (exp.f64 a)

   1. Initial program 88.7%

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

   3. Taylor expanded in b around 0 38.5%

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

   4. Taylor expanded in a around 0 36.6%

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

      1. *-commutative 36.6%

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

   6. Simplified 36.6%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 1:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.8% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in b around 0 65.8%

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

4. Taylor expanded in a around 0 39.0%

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

   1. *-commutative 39.0%

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

6. Simplified 39.0%

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

7. Final simplification 39.0%

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

Alternative 5: 39.7% 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 99.6%

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

3. Taylor expanded in a around 0 83.9%

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

4. Taylor expanded in b around 0 38.8%

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

5. Final simplification 38.8%

   \[\leadsto 0.5 \]
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 2024041 
(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))))