Quotient of sum of exps

Percentage Accurate: 98.9% → 99.1%

Time: 8.1s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ e^{a - \log \left(e^{a} + e^{b}\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (- a (log (+ (exp a) (exp b))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. add-exp-log 99.2%

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

   2. div-exp 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999995:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999995)
   (/ (exp a) (+ (exp a) 1.0))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999995) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.999995d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.999995) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.999995:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.999995)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.999995)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999995], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99999499999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 97.3%

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

   if 0.99999499999999997 < (exp.f64 a)

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 98.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999995:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(1 + e^{b - a}\right)}^{-3}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (cbrt (pow (+ 1.0 (exp (- b a))) -3.0)))

C

double code(double a, double b) {
	return cbrt(pow((1.0 + exp((b - a))), -3.0));
}

Java

public static double code(double a, double b) {
	return Math.cbrt(Math.pow((1.0 + Math.exp((b - a))), -3.0));
}

Julia

function code(a, b)
	return cbrt((Float64(1.0 + exp(Float64(b - a))) ^ -3.0))
end

Wolfram

code[a_, b_] := N[Power[N[Power[N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -3.0], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. add-cbrt-cube 98.6%

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

   2. pow1/3 99.2%

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

   3. pow3 99.2%

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

   4. clear-num 99.1%

      \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. inv-pow 99.1%

      \[\leadsto {\left({\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. pow-pow 99.1%

      \[\leadsto {\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

   7. metadata-eval 99.1%

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

3. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}} \]
4. Step-by-step derivation

   1. unpow1/3 98.6%

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

   2. +-commutative 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in b around inf 98.6%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}}^{-3}} \]
7. Step-by-step derivation

   1. rem-exp-log 98.6%

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

   2. exp-diff 99.0%

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

   3. sub-neg 99.0%

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

   4. +-commutative 99.0%

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

   5. prod-exp 98.6%

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

   6. rem-exp-log 98.6%

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

   7. distribute-lft-in 73.2%

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

   8. rec-exp 73.2%

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

   9. lft-mult-inverse 99.0%

      \[\leadsto \sqrt[3]{{\left(\color{blue}{1} + e^{-a} \cdot e^{b}\right)}^{-3}} \]

   10. prod-exp 99.4%

       \[\leadsto \sqrt[3]{{\left(1 + \color{blue}{e^{\left(-a\right) + b}}\right)}^{-3}} \]

8. Simplified 99.4%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(1 + e^{\left(-a\right) + b}\right)}}^{-3}} \]

9. Final simplification 99.4%

   \[\leadsto \sqrt[3]{{\left(1 + e^{b - a}\right)}^{-3}} \]

Alternative 4: 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 99.2%

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

2. Final simplification 99.2%

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

Alternative 5: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999995:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999995)
   (/ 1.0 (+ 1.0 (exp (- a))))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999995) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.999995d0) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.999995) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.999995:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.999995)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.999995)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999995], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99999499999999997

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. 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 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. pow3 99.9%

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

      4. clear-num 99.8%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. inv-pow 99.8%

         \[\leadsto {\left({\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. pow-pow 99.8%

         \[\leadsto {\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. unpow1/3 99.9%

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

      2. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in b around inf 99.9%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}}^{-3}} \]
   7. Step-by-step derivation

      1. rem-exp-log 99.9%

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

      2. exp-diff 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. prod-exp 99.9%

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

      6. rem-exp-log 99.9%

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

      7. distribute-lft-in 8.4%

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

      8. rec-exp 8.4%

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

      9. lft-mult-inverse 99.9%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{1} + e^{-a} \cdot e^{b}\right)}^{-3}} \]

      10. prod-exp 99.9%

          \[\leadsto \sqrt[3]{{\left(1 + \color{blue}{e^{\left(-a\right) + b}}\right)}^{-3}} \]

   8. Simplified 99.9%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(1 + e^{\left(-a\right) + b}\right)}}^{-3}} \]

   9. Taylor expanded in b around 0 97.3%

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

   if 0.99999499999999997 < (exp.f64 a)

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 98.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999995:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 6: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = exp(a);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	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[(N[Exp[b], $MachinePrecision] + 1.0), $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. Step-by-step derivation

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 97.2%

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

4. Final simplification 97.9%

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

Alternative 7: 71.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999996:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999996) (exp a) (/ 1.0 (+ 2.0 (+ b (* 0.5 (* b b)))))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999996) {
		tmp = exp(a);
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (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 (exp(a) <= 0.999999996d0) then
        tmp = exp(a)
    else
        tmp = 1.0d0 / (2.0d0 + (b + (0.5d0 * (b * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.999999996) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.999999996:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.999999996)
		tmp = exp(a);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b + Float64(0.5 * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.999999996)
		tmp = exp(a);
	else
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999999996], N[Exp[a], $MachinePrecision], N[(1.0 / N[(2.0 + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999999996000000002

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 93.5%

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

   if 0.999999996000000002 < (exp.f64 a)

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 98.7%

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

   3. Taylor expanded in b around 0 66.8%

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

      1. unpow2 66.8%

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

   5. Simplified 66.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999996:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]

Alternative 8: 57.4% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 3.5 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_0 - b}{b \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* 0.5 (* b b))))
   (if (<= b 3.5e+96)
     (/ 1.0 (- 2.0 a))
     (if (<= b 1.35e+154) (/ (- t_0 b) (* b t_0)) (/ -2.0 (* b b))))))

C

double code(double a, double b) {
	double t_0 = 0.5 * (b * b);
	double tmp;
	if (b <= 3.5e+96) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 1.35e+154) {
		tmp = (t_0 - b) / (b * t_0);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (b * b)
    if (b <= 3.5d+96) then
        tmp = 1.0d0 / (2.0d0 - a)
    else if (b <= 1.35d+154) then
        tmp = (t_0 - b) / (b * t_0)
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 0.5 * (b * b);
	double tmp;
	if (b <= 3.5e+96) {
		tmp = 1.0 / (2.0 - a);
	} else if (b <= 1.35e+154) {
		tmp = (t_0 - b) / (b * t_0);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 * (b * b)
	tmp = 0
	if b <= 3.5e+96:
		tmp = 1.0 / (2.0 - a)
	elif b <= 1.35e+154:
		tmp = (t_0 - b) / (b * t_0)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 * Float64(b * b))
	tmp = 0.0
	if (b <= 3.5e+96)
		tmp = Float64(1.0 / Float64(2.0 - a));
	elseif (b <= 1.35e+154)
		tmp = Float64(Float64(t_0 - b) / Float64(b * t_0));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 * (b * b);
	tmp = 0.0;
	if (b <= 3.5e+96)
		tmp = 1.0 / (2.0 - a);
	elseif (b <= 1.35e+154)
		tmp = (t_0 - b) / (b * t_0);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.5e+96], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(N[(t$95$0 - b), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.4999999999999999e96

   1. Initial program 99.5%

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

      1. add-cbrt-cube 98.8%

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

      2. pow1/3 99.4%

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

      3. pow3 99.4%

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

      4. clear-num 99.4%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. inv-pow 99.4%

         \[\leadsto {\left({\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. pow-pow 99.4%

         \[\leadsto {\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. unpow1/3 98.8%

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

      2. +-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in b around inf 98.8%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}}^{-3}} \]
   7. Step-by-step derivation

      1. rem-exp-log 98.8%

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

      2. exp-diff 98.8%

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

      3. sub-neg 98.8%

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

      4. +-commutative 98.8%

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

      5. prod-exp 98.8%

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

      6. rem-exp-log 98.8%

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

      7. distribute-lft-in 75.3%

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

      8. rec-exp 75.3%

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

      9. lft-mult-inverse 99.3%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{1} + e^{-a} \cdot e^{b}\right)}^{-3}} \]

      10. prod-exp 99.3%

          \[\leadsto \sqrt[3]{{\left(1 + \color{blue}{e^{\left(-a\right) + b}}\right)}^{-3}} \]

   8. Simplified 99.3%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(1 + e^{\left(-a\right) + b}\right)}}^{-3}} \]

   9. Taylor expanded in b around 0 75.5%

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

   10. Taylor expanded in a around 0 50.7%

       \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot a}} \]
   11. Step-by-step derivation

       1. mul-1-neg 50.7%

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

       2. unsub-neg 50.7%

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

   12. Simplified 50.7%

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

   if 3.4999999999999999e96 < b < 1.35000000000000003e154

   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. Taylor expanded in b around 0 4.3%

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

      1. +-commutative 4.3%

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

   5. Simplified 4.3%

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

   6. Taylor expanded in b around inf 4.3%

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

      1. associate-*r/ 4.3%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 4.3%

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

      3. unpow2 4.3%

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

   8. Simplified 4.3%

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

      1. clear-num 4.3%

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

      2. frac-sub 90.4%

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

      3. *-un-lft-identity 90.4%

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

      4. div-inv 90.4%

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

      5. metadata-eval 90.4%

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

      6. metadata-eval 90.4%

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

      7. div-inv 90.4%

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

      8. /-rgt-identity 90.4%

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

      9. div-inv 90.4%

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

      10. metadata-eval 90.4%

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

   10. Applied egg-rr 90.4%

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

   if 1.35000000000000003e154 < b

   1. Initial program 97.3%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 7.2%

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

      1. +-commutative 7.2%

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

   5. Simplified 7.2%

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

   6. Taylor expanded in b around inf 7.2%

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

      1. associate-*r/ 7.2%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 7.2%

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

      3. unpow2 7.2%

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

   8. Simplified 7.2%

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

   9. Taylor expanded in b around 0 100.0%

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

       1. unpow2 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5 \cdot \left(b \cdot b\right) - b}{b \cdot \left(0.5 \cdot \left(b \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 9: 53.0% accurate, 23.3× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (2.0 + (b + (0.5 * (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 = 1.0d0 / (2.0d0 + (b + (0.5d0 * (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 = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.0:
		tmp = 0.5
	else:
		tmp = 1.0 / (2.0 + (b + (0.5 * (b * b))))
	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(0.5 * 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 = 1.0 / (2.0 + (b + (0.5 * (b * b))));
	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[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 0 100.0%

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

   3. Taylor expanded in b around 0 18.8%

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

   if -2 < b

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0 79.0%

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

   3. Taylor expanded in b around 0 63.8%

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

      1. unpow2 63.8%

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

   5. Simplified 63.8%

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

4. Final simplification 56.6%

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

Alternative 10: 52.6% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0108:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 0.0108) (+ 0.5 (* a 0.25)) (/ -2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 0.0108) {
		tmp = 0.5 + (a * 0.25);
	} 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 <= 0.0108d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    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 <= 0.0108) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 0.0108:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 0.0108)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 0.0108)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 0.0108], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.010800000000000001

   1. Initial program 99.4%

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

   2. Taylor expanded in b around 0 79.7%

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

   3. Taylor expanded in a around 0 56.4%

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

      1. *-commutative 56.4%

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

   5. Simplified 56.4%

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

   if 0.010800000000000001 < b

   1. Initial program 98.6%

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

   2. Taylor expanded in a around 0 98.6%

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

   3. Taylor expanded in b around 0 5.6%

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

      1. +-commutative 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in b around inf 5.6%

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

      1. associate-*r/ 5.6%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 5.6%

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

      3. unpow2 5.6%

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

   8. Simplified 5.6%

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

   9. Taylor expanded in b around 0 54.6%

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

       1. unpow2 54.6%

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

   11. Simplified 54.6%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0108:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 11: 53.1% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.35 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.35e+72) (/ 1.0 (- 2.0 a)) (/ -2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.35e+72) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.35d+72) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = (-2.0d0) / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.35e+72) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = -2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.35e+72:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = -2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.35e+72)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(-2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.35e+72)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = -2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.35e+72], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.3499999999999999e72

   1. Initial program 99.5%

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

      1. add-cbrt-cube 98.8%

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

      2. pow1/3 99.4%

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

      3. pow3 99.4%

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

      4. clear-num 99.4%

         \[\leadsto {\left({\color{blue}{\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. inv-pow 99.4%

         \[\leadsto {\left({\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. pow-pow 99.4%

         \[\leadsto {\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}} \]
   4. Step-by-step derivation

      1. unpow1/3 98.8%

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

      2. +-commutative 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in b around inf 98.8%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}}^{-3}} \]
   7. Step-by-step derivation

      1. rem-exp-log 98.8%

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

      2. exp-diff 98.8%

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

      3. sub-neg 98.8%

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

      4. +-commutative 98.8%

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

      5. prod-exp 98.8%

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

      6. rem-exp-log 98.8%

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

      7. distribute-lft-in 75.5%

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

      8. rec-exp 75.5%

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

      9. lft-mult-inverse 99.2%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{1} + e^{-a} \cdot e^{b}\right)}^{-3}} \]

      10. prod-exp 99.2%

          \[\leadsto \sqrt[3]{{\left(1 + \color{blue}{e^{\left(-a\right) + b}}\right)}^{-3}} \]

   8. Simplified 99.2%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(1 + e^{\left(-a\right) + b}\right)}}^{-3}} \]

   9. Taylor expanded in b around 0 77.1%

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

   10. Taylor expanded in a around 0 52.3%

       \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot a}} \]
   11. Step-by-step derivation

       1. mul-1-neg 52.3%

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

       2. unsub-neg 52.3%

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

   12. Simplified 52.3%

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

   if 3.3499999999999999e72 < b

   1. Initial program 98.1%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 6.2%

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

      1. +-commutative 6.2%

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

   5. Simplified 6.2%

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

   6. Taylor expanded in b around inf 6.2%

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

      1. associate-*r/ 6.2%

         \[\leadsto \frac{1}{b} - \color{blue}{\frac{2 \cdot 1}{{b}^{2}}} \]

      2. metadata-eval 6.2%

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

      3. unpow2 6.2%

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

   8. Simplified 6.2%

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

   9. Taylor expanded in b around 0 70.6%

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

       1. unpow2 70.6%

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

   11. Simplified 70.6%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.35 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]

Alternative 12: 39.1% 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 99.2%

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

2. Taylor expanded in b around 0 67.9%

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

3. Taylor expanded in a around 0 41.5%

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

   1. *-commutative 41.5%

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

5. Simplified 41.5%

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

6. Final simplification 41.5%

   \[\leadsto 0.5 + a \cdot 0.25 \]

Alternative 13: 38.9% 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 99.2%

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

2. Taylor expanded in a around 0 82.4%

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

3. Taylor expanded in b around 0 41.4%

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

4. Final simplification 41.4%

   \[\leadsto 0.5 \]

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 2023238 
(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))))