Quotient of sum of exps

Percentage Accurate: 99.0% → 99.0%

Time: 6.0s

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: 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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

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 100.0%

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

2. Final simplification 100.0%

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

Alternative 3: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9995:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9995) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9995) {
		tmp = 1.0 / (1.0 + 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) <= 0.9995d0) then
        tmp = 1.0d0 / (1.0d0 + 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) <= 0.9995) {
		tmp = 1.0 / (1.0 + 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) <= 0.9995:
		tmp = 1.0 / (1.0 + 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) <= 0.9995)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-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) <= 0.9995)
		tmp = 1.0 / (1.0 + 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], 0.9995], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $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) < 0.99950000000000006

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

      1. remove-double-div 100.0%

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

      2. exp-neg 100.0%

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

      3. associate-/r* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-lft-in 2.8%

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

      6. exp-neg 2.8%

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

      7. lft-mult-inverse 100.0%

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

      8. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   if 0.99950000000000006 < (exp.f64 a)

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9995:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 4: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9995:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9995) (/ (exp a) 2.0) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9995) {
		tmp = exp(a) / 2.0;
	} 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) <= 0.9995d0) then
        tmp = exp(a) / 2.0d0
    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) <= 0.9995) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.9995:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.9995)
		tmp = Float64(exp(a) / 2.0);
	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) <= 0.9995)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.9995], N[(N[Exp[a], $MachinePrecision] / 2.0), $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) < 0.99950000000000006

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 97.9%

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

   if 0.99950000000000006 < (exp.f64 a)

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9995:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 5: 80.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -31:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -31.0) 1.0 (/ (exp a) 2.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -31.0) {
		tmp = 1.0;
	} else {
		tmp = exp(a) / 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 <= (-31.0d0)) then
        tmp = 1.0d0
    else
        tmp = exp(a) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -31.0) {
		tmp = 1.0;
	} else {
		tmp = Math.exp(a) / 2.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -31.0:
		tmp = 1.0
	else:
		tmp = math.exp(a) / 2.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -31.0)
		tmp = 1.0;
	else
		tmp = Float64(exp(a) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -31.0)
		tmp = 1.0;
	else
		tmp = exp(a) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -31.0], 1.0, N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -31

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

         \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

   4. Applied egg-rr 100.0%

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

      1. add-sqr-sqrt 100.0%

         \[\leadsto \color{blue}{\sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}}} \]

      2. sqrt-unprod 100.0%

         \[\leadsto \color{blue}{\sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}}} \]

      3. exp-neg 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{b}\right)}}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\left(-1 + 1\right)}}} \]

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1} \]

   if -31 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 79.4%

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

   3. Taylor expanded in a around 0 78.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -31:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \]

Alternative 6: 54.3% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.92:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.92) 1.0 (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.92) {
		tmp = 1.0;
	} 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 (b <= (-0.92d0)) then
        tmp = 1.0d0
    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 (b <= -0.92) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.92:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.92)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.92)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.92], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.92000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 98.0%

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

      1. add-exp-log 98.0%

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

      2. log-rec 98.0%

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

      3. log1p-udef 98.0%

         \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

   4. Applied egg-rr 98.0%

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

      1. add-sqr-sqrt 98.0%

         \[\leadsto \color{blue}{\sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}}} \]

      2. sqrt-unprod 98.0%

         \[\leadsto \color{blue}{\sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}}} \]

      3. exp-neg 98.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{b}\right)}}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      4. inv-pow 98.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 98.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 98.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 98.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 98.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 97.9%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 98.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 98.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 98.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 98.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 98.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\left(-1 + 1\right)}}} \]

      15. metadata-eval 98.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   6. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{1} \]

   if -0.92000000000000004 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 79.3%

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

   3. Taylor expanded in a around 0 46.8%

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

      1. *-commutative 46.8%

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

   5. Simplified 46.8%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.92:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 7: 54.2% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -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 (b <= (-2.0d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (b * (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.0:
		tmp = 1.0
	else:
		tmp = 0.5 + (b * -0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(b * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -2.0)
		tmp = 1.0;
	else
		tmp = 0.5 + (b * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2.0], 1.0, N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 98.0%

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

      1. add-exp-log 98.0%

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

      2. log-rec 98.0%

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

      3. log1p-udef 98.0%

         \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

   4. Applied egg-rr 98.0%

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

      1. add-sqr-sqrt 98.0%

         \[\leadsto \color{blue}{\sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}}} \]

      2. sqrt-unprod 98.0%

         \[\leadsto \color{blue}{\sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}}} \]

      3. exp-neg 98.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{b}\right)}}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      4. inv-pow 98.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 98.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 98.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 98.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 98.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 97.9%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 98.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 98.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 98.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 98.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 98.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\left(-1 + 1\right)}}} \]

      15. metadata-eval 98.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   6. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{1} \]

   if -2 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 77.4%

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

   3. Taylor expanded in b around 0 47.4%

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

      1. *-commutative 47.4%

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

   5. Simplified 47.4%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]

Alternative 8: 54.1% accurate, 99.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -31:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -31.0) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -31.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-31.0d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -31.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -31.0:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -31.0)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -31.0)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -31.0], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -31

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

         \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]

   4. Applied egg-rr 100.0%

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

      1. add-sqr-sqrt 100.0%

         \[\leadsto \color{blue}{\sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}} \cdot \sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)}}} \]

      2. sqrt-unprod 100.0%

         \[\leadsto \color{blue}{\sqrt{e^{-\mathsf{log1p}\left(e^{b}\right)} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}}} \]

      3. exp-neg 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{b}\right)}}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      4. inv-pow 100.0%

         \[\leadsto \sqrt{\color{blue}{{\left(e^{\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      5. pow-exp 100.0%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(e^{b}\right) \cdot -1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      7. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b}\right) \cdot \mathsf{log1p}\left(e^{b}\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      8. sqr-neg 100.0%

         \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b}\right)\right)}} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      9. sqrt-unprod 100.0%

         \[\leadsto \sqrt{e^{\color{blue}{\left(\sqrt{-\mathsf{log1p}\left(e^{b}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      10. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{b}\right)\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      11. add-log-exp 100.0%

          \[\leadsto \sqrt{e^{\color{blue}{\log \left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)} \cdot -1} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      12. pow-to-exp 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1}} \cdot e^{-\mathsf{log1p}\left(e^{b}\right)}} \]

      13. pow1 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{-1} \cdot \color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{1}}} \]

      14. pow-prod-up 100.0%

          \[\leadsto \sqrt{\color{blue}{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\left(-1 + 1\right)}}} \]

      15. metadata-eval 100.0%

          \[\leadsto \sqrt{{\left(e^{-\mathsf{log1p}\left(e^{b}\right)}\right)}^{\color{blue}{0}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1} \]

   if -31 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 77.1%

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

   3. Taylor expanded in b around 0 46.3%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -31:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 39.6% 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 100.0%

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

2. Taylor expanded in a around 0 81.3%

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

3. Taylor expanded in b around 0 41.3%

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

4. Final simplification 41.3%

   \[\leadsto 0.5 \]

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 2023331 
(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))))