Quotient of sum of exps

Percentage Accurate: 98.9% → 98.3%

Time: 11.5s

Alternatives: 17

Speedup: 1.0×

• could not determine a ground truth

  1. a = 6.357281191345618e+174
  2. b = -1.816778967687896e-192

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.998:\\ \;\;\;\;\frac{1}{e^{-a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.998) (/ 1.0 (+ (exp (- a)) 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.998) {
		tmp = 1.0 / (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) <= 0.998d0) then
        tmp = 1.0d0 / (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) <= 0.998) {
		tmp = 1.0 / (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) <= 0.998:
		tmp = 1.0 / (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) <= 0.998)
		tmp = Float64(1.0 / Float64(exp(Float64(-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) <= 0.998)
		tmp = 1.0 / (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], 0.998], N[(1.0 / 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) < 0.998

   1. Initial program 96.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 96.9

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

         \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. lower-neg.f64 98.4

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

   7. Simplified 98.4%

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

   if 0.998 < (exp.f64 a)

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.998:\\ \;\;\;\;\frac{1}{e^{-a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (fma (log (+ (exp a) (exp b))) -1.0 a)))

C

double code(double a, double b) {
	return exp(fma(log((exp(a) + exp(b))), -1.0, a));
}

Julia

function code(a, b)
	return exp(fma(log(Float64(exp(a) + exp(b))), -1.0, a))
end

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. inv-pow N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. prod-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-log.f64 98.8

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

4. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

Alternative 4: 97.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+24}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.2e+24) (* (exp a) 0.5) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.2e+24) {
		tmp = exp(a) * 0.5;
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-4.2d+24)) then
        tmp = exp(a) * 0.5d0
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.2e+24) {
		tmp = Math.exp(a) * 0.5;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.2e+24:
		tmp = math.exp(a) * 0.5
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.2e+24)
		tmp = Float64(exp(a) * 0.5);
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.2e+24)
		tmp = exp(a) * 0.5;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.2e+24], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.2000000000000003e24

   1. Initial program 98.3%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{2}} \]
   9. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. div-inv N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      3. metadata-eval N/A

         \[\leadsto e^{a} \cdot \color{blue}{\frac{1}{2}} \]

      4. lower-*.f64 100.0

         \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   if -4.2000000000000003e24 < a

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 98.7

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

   5. Simplified 98.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+24}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq -7400:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+32}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot \left(t\_0 \cdot t\_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b -7400.0)
     (+ (exp b) 1.0)
     (if (<= b 1.3e+32)
       (* (exp a) 0.5)
       (/ 1.0 (* b (* b (* b (* t_0 t_0)))))))))

C

double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= -7400.0) {
		tmp = exp(b) + 1.0;
	} else if (b <= 1.3e+32) {
		tmp = exp(a) * 0.5;
	} else {
		tmp = 1.0 / (b * (b * (b * (t_0 * t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * b)
    if (b <= (-7400.0d0)) then
        tmp = exp(b) + 1.0d0
    else if (b <= 1.3d+32) then
        tmp = exp(a) * 0.5d0
    else
        tmp = 1.0d0 / (b * (b * (b * (t_0 * t_0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= -7400.0) {
		tmp = Math.exp(b) + 1.0;
	} else if (b <= 1.3e+32) {
		tmp = Math.exp(a) * 0.5;
	} else {
		tmp = 1.0 / (b * (b * (b * (t_0 * t_0))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * b)
	tmp = 0
	if b <= -7400.0:
		tmp = math.exp(b) + 1.0
	elif b <= 1.3e+32:
		tmp = math.exp(a) * 0.5
	else:
		tmp = 1.0 / (b * (b * (b * (t_0 * t_0))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= -7400.0)
		tmp = Float64(exp(b) + 1.0);
	elseif (b <= 1.3e+32)
		tmp = Float64(exp(a) * 0.5);
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * Float64(b * Float64(t_0 * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * b);
	tmp = 0.0;
	if (b <= -7400.0)
		tmp = exp(b) + 1.0;
	elseif (b <= 1.3e+32)
		tmp = exp(a) * 0.5;
	else
		tmp = 1.0 / (b * (b * (b * (t_0 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -7400.0], N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[b, 1.3e+32], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -7400

   1. Initial program 98.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -7400 < b < 1.3000000000000001e32

   1. Initial program 98.5%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 95.1%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 93.9%

      \[\leadsto \frac{e^{a}}{\color{blue}{2}} \]
   9. Step-by-step derivation

      1. lift-exp.f64 N/A

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

      2. div-inv N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      3. metadata-eval N/A

         \[\leadsto e^{a} \cdot \color{blue}{\frac{1}{2}} \]

      4. lower-*.f64 93.9

         \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   10. Applied egg-rr 93.9%

       \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   if 1.3000000000000001e32 < b

   1. Initial program 97.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.6

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

   8. Simplified 5.6%

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

      1. flip3-+ N/A

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

      2. associate-/r/ N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   10. Applied egg-rr 7.6%

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

   11. Taylor expanded in b around -inf

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{{b}^{9}}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{{b}^{\color{blue}{\left(8 + 1\right)}}} \]

       3. pow-plus N/A

          \[\leadsto \frac{1}{\color{blue}{{b}^{8} \cdot b}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{{b}^{8} \cdot b}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{1}{{b}^{\color{blue}{\left(7 + 1\right)}} \cdot b} \]

       6. pow-plus N/A

          \[\leadsto \frac{1}{\color{blue}{\left({b}^{7} \cdot b\right)} \cdot b} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left({b}^{7} \cdot b\right)} \cdot b} \]

       8. metadata-eval N/A

          \[\leadsto \frac{1}{\left({b}^{\color{blue}{\left(6 + 1\right)}} \cdot b\right) \cdot b} \]

       9. pow-plus N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left({b}^{6} \cdot b\right)} \cdot b\right) \cdot b} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\left(\color{blue}{\left(b \cdot {b}^{6}\right)} \cdot b\right) \cdot b} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{1}{\left(\color{blue}{\left(b \cdot {b}^{6}\right)} \cdot b\right) \cdot b} \]

       12. metadata-eval N/A

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

       13. pow-sqr N/A

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

       14. lower-*.f64 N/A

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

       15. cube-mult N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 N/A

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

       20. cube-mult N/A

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

       21. unpow2 N/A

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

       22. lower-*.f64 N/A

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

       23. unpow2 N/A

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

       24. lower-*.f64 98.7

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

   13. Simplified 98.7%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7400:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+32}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq -7400:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot \left(t\_0 \cdot t\_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b -7400.0)
     (+ (exp b) 1.0)
     (if (<= b 6.5e+18)
       (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
       (/ 1.0 (* b (* b (* b (* t_0 t_0)))))))))

C

double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= -7400.0) {
		tmp = exp(b) + 1.0;
	} else if (b <= 6.5e+18) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / (b * (b * (b * (t_0 * t_0))));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= -7400.0)
		tmp = Float64(exp(b) + 1.0);
	elseif (b <= 6.5e+18)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * Float64(b * Float64(t_0 * t_0)))));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -7400.0], N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[b, 6.5e+18], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -7400

   1. Initial program 98.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -7400 < b < 6.5e18

   1. Initial program 98.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

         \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. lower-neg.f64 96.5

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

   7. Simplified 96.5%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1, 2\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \color{blue}{-1}, 2\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{-1}{6} \cdot a, -1\right)}, 2\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{-1}{6} \cdot a + \frac{1}{2}}, -1\right), 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 2\right)} \]

      8. lower-fma.f64 86.3

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, -0.16666666666666666, 0.5\right)}, -1\right), 2\right)} \]

   10. Simplified 86.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}} \]

   if 6.5e18 < b

   1. Initial program 97.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.5

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

   8. Simplified 5.5%

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

      1. flip3-+ N/A

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

      2. associate-/r/ N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   10. Applied egg-rr 7.5%

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

   11. Taylor expanded in b around -inf

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{{b}^{9}}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{{b}^{\color{blue}{\left(8 + 1\right)}}} \]

       3. pow-plus N/A

          \[\leadsto \frac{1}{\color{blue}{{b}^{8} \cdot b}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{{b}^{8} \cdot b}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{1}{{b}^{\color{blue}{\left(7 + 1\right)}} \cdot b} \]

       6. pow-plus N/A

          \[\leadsto \frac{1}{\color{blue}{\left({b}^{7} \cdot b\right)} \cdot b} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left({b}^{7} \cdot b\right)} \cdot b} \]

       8. metadata-eval N/A

          \[\leadsto \frac{1}{\left({b}^{\color{blue}{\left(6 + 1\right)}} \cdot b\right) \cdot b} \]

       9. pow-plus N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left({b}^{6} \cdot b\right)} \cdot b\right) \cdot b} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\left(\color{blue}{\left(b \cdot {b}^{6}\right)} \cdot b\right) \cdot b} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{1}{\left(\color{blue}{\left(b \cdot {b}^{6}\right)} \cdot b\right) \cdot b} \]

       12. metadata-eval N/A

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

       13. pow-sqr N/A

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

       14. lower-*.f64 N/A

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

       15. cube-mult N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 N/A

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

       20. cube-mult N/A

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

       21. unpow2 N/A

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

       22. lower-*.f64 N/A

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

       23. unpow2 N/A

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

       24. lower-*.f64 96.0

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

   13. Simplified 96.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7400:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot \left(t\_0 \cdot t\_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b 6.5e+18)
     (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
     (/ 1.0 (* b (* b (* b (* t_0 t_0))))))))

C

double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= 6.5e+18) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / (b * (b * (b * (t_0 * t_0))));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 6.5e+18)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * Float64(b * Float64(t_0 * t_0)))));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 6.5e+18], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5e18

   1. Initial program 98.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.4

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

         \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. lower-neg.f64 75.5

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

   7. Simplified 75.5%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1, 2\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \color{blue}{-1}, 2\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{-1}{6} \cdot a, -1\right)}, 2\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{-1}{6} \cdot a + \frac{1}{2}}, -1\right), 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 2\right)} \]

      8. lower-fma.f64 68.1

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, -0.16666666666666666, 0.5\right)}, -1\right), 2\right)} \]

   10. Simplified 68.1%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}} \]

   if 6.5e18 < b

   1. Initial program 97.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.5

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

   8. Simplified 5.5%

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

      1. flip3-+ N/A

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

      2. associate-/r/ N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   10. Applied egg-rr 7.5%

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

   11. Taylor expanded in b around -inf

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{{b}^{9}}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{{b}^{\color{blue}{\left(8 + 1\right)}}} \]

       3. pow-plus N/A

          \[\leadsto \frac{1}{\color{blue}{{b}^{8} \cdot b}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{{b}^{8} \cdot b}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{1}{{b}^{\color{blue}{\left(7 + 1\right)}} \cdot b} \]

       6. pow-plus N/A

          \[\leadsto \frac{1}{\color{blue}{\left({b}^{7} \cdot b\right)} \cdot b} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left({b}^{7} \cdot b\right)} \cdot b} \]

       8. metadata-eval N/A

          \[\leadsto \frac{1}{\left({b}^{\color{blue}{\left(6 + 1\right)}} \cdot b\right) \cdot b} \]

       9. pow-plus N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left({b}^{6} \cdot b\right)} \cdot b\right) \cdot b} \]

       10. *-commutative N/A

           \[\leadsto \frac{1}{\left(\color{blue}{\left(b \cdot {b}^{6}\right)} \cdot b\right) \cdot b} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{1}{\left(\color{blue}{\left(b \cdot {b}^{6}\right)} \cdot b\right) \cdot b} \]

       12. metadata-eval N/A

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

       13. pow-sqr N/A

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

       14. lower-*.f64 N/A

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

       15. cube-mult N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 N/A

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

       20. cube-mult N/A

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

       21. unpow2 N/A

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

       22. lower-*.f64 N/A

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

       23. unpow2 N/A

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

       24. lower-*.f64 96.0

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

   13. Simplified 96.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(t\_0 \cdot t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b 6.5e+18)
     (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
     (/ 1.0 (* b (* b (* t_0 t_0)))))))

C

double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= 6.5e+18) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / (b * (b * (t_0 * t_0)));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 6.5e+18)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * Float64(t_0 * t_0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 6.5e+18], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5e18

   1. Initial program 98.4%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.4

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

         \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. lower-neg.f64 75.5

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

   7. Simplified 75.5%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1, 2\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \color{blue}{-1}, 2\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{-1}{6} \cdot a, -1\right)}, 2\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{-1}{6} \cdot a + \frac{1}{2}}, -1\right), 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 2\right)} \]

      8. lower-fma.f64 68.1

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, -0.16666666666666666, 0.5\right)}, -1\right), 2\right)} \]

   10. Simplified 68.1%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}} \]

   if 6.5e18 < b

   1. Initial program 97.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.5

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

   8. Simplified 5.5%

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

      1. flip3-+ N/A

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

      2. associate-/r/ N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   10. Applied egg-rr 7.5%

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

   11. Taylor expanded in b around inf

       \[\leadsto \color{blue}{\frac{1}{{b}^{8}}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{{b}^{8}}} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1}{{b}^{\color{blue}{\left(7 + 1\right)}}} \]

       3. pow-plus N/A

          \[\leadsto \frac{1}{\color{blue}{{b}^{7} \cdot b}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{{b}^{7} \cdot b}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{1}{{b}^{\color{blue}{\left(6 + 1\right)}} \cdot b} \]

       6. pow-plus N/A

          \[\leadsto \frac{1}{\color{blue}{\left({b}^{6} \cdot b\right)} \cdot b} \]

       7. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{\left(b \cdot {b}^{6}\right)} \cdot b} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(b \cdot {b}^{6}\right)} \cdot b} \]

       9. metadata-eval N/A

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

       10. pow-sqr N/A

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

       11. lower-*.f64 N/A

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

       12. cube-mult N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 N/A

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

       15. unpow2 N/A

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

       16. lower-*.f64 N/A

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

       17. cube-mult N/A

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

       18. unpow2 N/A

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

       19. lower-*.f64 N/A

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

       20. unpow2 N/A

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

       21. lower-*.f64 96.0

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

   13. Simplified 96.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-16}{\mathsf{fma}\left(b, b \cdot b, 8\right) \cdot \left(b \cdot b + \left(b \cdot 2 - 4\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.6e+59)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (/ -16.0 (* (fma b (* b b) 8.0) (+ (* b b) (- (* b 2.0) 4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.6e+59) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = -16.0 / (fma(b, (b * b), 8.0) * ((b * b) + ((b * 2.0) - 4.0)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.6e+59)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(-16.0 / Float64(fma(b, Float64(b * b), 8.0) * Float64(Float64(b * b) + Float64(Float64(b * 2.0) - 4.0))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.6e+59], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(-16.0 / N[(N[(b * N[(b * b), $MachinePrecision] + 8.0), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] + N[(N[(b * 2.0), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.6000000000000002e59

   1. Initial program 97.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

         \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. lower-neg.f64 74.3

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

   7. Simplified 74.3%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1, 2\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \color{blue}{-1}, 2\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{-1}{6} \cdot a, -1\right)}, 2\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{-1}{6} \cdot a + \frac{1}{2}}, -1\right), 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 2\right)} \]

      8. lower-fma.f64 66.6

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, -0.16666666666666666, 0.5\right)}, -1\right), 2\right)} \]

   10. Simplified 66.6%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}} \]

   if 7.6000000000000002e59 < b

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 5.7

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

   8. Simplified 5.7%

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

      1. flip3-+ N/A

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

      2. associate-/r/ N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

   10. Applied egg-rr 7.9%

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

   11. Taylor expanded in b around 0

       \[\leadsto \frac{\color{blue}{-16}}{\mathsf{fma}\left(b, b \cdot b, 8\right) \cdot \left(b \cdot b - \left(4 - b \cdot 2\right)\right)} \]
   12. Step-by-step derivation

   13. Simplified 98.6%

       \[\leadsto \frac{\color{blue}{-16}}{\mathsf{fma}\left(b, b \cdot b, 8\right) \cdot \left(b \cdot b - \left(4 - b \cdot 2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-16}{\mathsf{fma}\left(b, b \cdot b, 8\right) \cdot \left(b \cdot b + \left(b \cdot 2 - 4\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.08 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.08e+80)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.08e+80) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.08e+80)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.08e+80], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.08e80

   1. Initial program 98.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

         \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. lower-neg.f64 73.2

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

   7. Simplified 73.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1, 2\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \color{blue}{-1}, 2\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{-1}{6} \cdot a, -1\right)}, 2\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{-1}{6} \cdot a + \frac{1}{2}}, -1\right), 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 2\right)} \]

      8. lower-fma.f64 65.8

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, -0.16666666666666666, 0.5\right)}, -1\right), 2\right)} \]

   10. Simplified 65.8%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}} \]

   if 1.08e80 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, 1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), 2\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{1}{6} \cdot b, 1\right)}, 2\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, 1\right), 2\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 2\right)} \]

      7. lower-fma.f64 95.1

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)}, 1\right), 2\right)} \]

   8. Simplified 95.1%

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

Alternative 11: 67.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.45 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.45e+148)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (/ 1.0 (fma b (fma b 0.5 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.45e+148) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.45e+148)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.45e+148], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.45e148

   1. Initial program 98.1%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.6

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

         \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. lower-neg.f64 69.2

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

   7. Simplified 69.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right) + 2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1, 2\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) + \color{blue}{-1}, 2\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{-1}{6} \cdot a, -1\right)}, 2\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{-1}{6} \cdot a + \frac{1}{2}}, -1\right), 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 2\right)} \]

      8. lower-fma.f64 61.9

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, -0.16666666666666666, 0.5\right)}, -1\right), 2\right)} \]

   10. Simplified 61.9%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}} \]

   if 2.45e148 < b

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 95.8

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 2\right)} \]

   8. Simplified 95.8%

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

Alternative 12: 64.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.45 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a, -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.45e+148)
   (/ 1.0 (fma a (fma 0.5 a -1.0) 2.0))
   (/ 1.0 (fma b (fma b 0.5 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.45e+148) {
		tmp = 1.0 / fma(a, fma(0.5, a, -1.0), 2.0);
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.45e+148)
		tmp = Float64(1.0 / fma(a, fma(0.5, a, -1.0), 2.0));
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.45e+148], N[(1.0 / N[(a * N[(0.5 * a + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.45e148

   1. Initial program 98.1%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.6

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

         \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. lower-neg.f64 69.2

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

   7. Simplified 69.2%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \frac{1}{2} \cdot a - 1, 2\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)} \]

      4. metadata-eval N/A

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

      5. lower-fma.f64 58.7

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(0.5, a, -1\right)}, 2\right)} \]

   10. Simplified 58.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a, -1\right), 2\right)}} \]

   if 2.45e148 < b

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 95.8

         \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, 0.5, 1\right)}, 2\right)} \]

   8. Simplified 95.8%

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

Alternative 13: 64.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a, -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + -2}{\mathsf{fma}\left(b, b, -4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.5e+148)
   (/ 1.0 (fma a (fma 0.5 a -1.0) 2.0))
   (/ (+ b -2.0) (fma b b -4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.5e+148) {
		tmp = 1.0 / fma(a, fma(0.5, a, -1.0), 2.0);
	} else {
		tmp = (b + -2.0) / fma(b, b, -4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.5e+148)
		tmp = Float64(1.0 / fma(a, fma(0.5, a, -1.0), 2.0));
	else
		tmp = Float64(Float64(b + -2.0) / fma(b, b, -4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.5e+148], N[(1.0 / N[(a * N[(0.5 * a + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(b + -2.0), $MachinePrecision] / N[(b * b + -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.50000000000000012e148

   1. Initial program 98.1%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.6

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

         \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. lower-neg.f64 69.2

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

   7. Simplified 69.2%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \frac{1}{2} \cdot a - 1, 2\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\frac{1}{2} \cdot a + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)} \]

      4. metadata-eval N/A

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

      5. lower-fma.f64 58.7

         \[\leadsto \frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(0.5, a, -1\right)}, 2\right)} \]

   10. Simplified 58.7%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(0.5, a, -1\right), 2\right)}} \]

   if 2.50000000000000012e148 < b

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 6.5

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

   8. Simplified 6.5%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. sub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \frac{b + -2}{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{neg}\left(2 \cdot 2\right)\right)}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{b + -2}{\mathsf{fma}\left(b, b, \mathsf{neg}\left(\color{blue}{4}\right)\right)} \]

      12. metadata-eval 95.5

          \[\leadsto \frac{b + -2}{\mathsf{fma}\left(b, b, \color{blue}{-4}\right)} \]

   10. Applied egg-rr 95.5%

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

Alternative 14: 53.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + -2}{\mathsf{fma}\left(b, b, -4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4e-64) (/ 1.0 (- 2.0 a)) (/ (+ b -2.0) (fma b b -4.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4e-64) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = (b + -2.0) / fma(b, b, -4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4e-64)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(Float64(b + -2.0) / fma(b, b, -4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4e-64], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(N[(b + -2.0), $MachinePrecision] / N[(b * b + -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.99999999999999986e-64

   1. Initial program 98.8%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. inv-pow N/A

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

      8. pow-to-exp N/A

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

      9. lift-exp.f64 N/A

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

      10. prod-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-log.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

         \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

      2. unsub-neg N/A

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

      3. exp-diff N/A

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

      4. remove-double-div N/A

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

      5. exp-neg N/A

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

      6. rem-exp-log N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. exp-neg N/A

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

      12. lft-mult-inverse N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lower-exp.f64 N/A

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

      17. neg-mul-1 N/A

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

      18. lower-neg.f64 74.5

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

   7. Simplified 74.5%

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

   8. Taylor expanded in a around 0

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

      1. neg-mul-1 N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 52.1

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

   10. Simplified 52.1%

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

   if 3.99999999999999986e-64 < b

   1. Initial program 96.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 99.0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 17.7

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

   8. Simplified 17.7%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f64 N/A

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

      8. sub-neg N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \frac{b + -2}{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{neg}\left(2 \cdot 2\right)\right)}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{b + -2}{\mathsf{fma}\left(b, b, \mathsf{neg}\left(\color{blue}{4}\right)\right)} \]

      12. metadata-eval 61.1

          \[\leadsto \frac{b + -2}{\mathsf{fma}\left(b, b, \color{blue}{-4}\right)} \]

   10. Applied egg-rr 61.1%

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

Alternative 15: 40.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (- 2.0 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 - a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 - a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 - a);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. inv-pow N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. prod-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-log.f64 98.8

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

      \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

   4. remove-double-div N/A

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

   5. exp-neg N/A

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

   6. rem-exp-log N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. exp-neg N/A

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

   12. lft-mult-inverse N/A

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

   13. *-rgt-identity N/A

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

   14. lower-+.f64 N/A

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

   15. neg-mul-1 N/A

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

   16. lower-exp.f64 N/A

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

   17. neg-mul-1 N/A

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

   18. lower-neg.f64 62.9

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

7. Simplified 62.9%

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

8. Taylor expanded in a around 0

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

   1. neg-mul-1 N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 39.9

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

10. Simplified 39.9%

    \[\leadsto \frac{1}{\color{blue}{2 - a}} \]
11. Add Preprocessing

Alternative 16: 39.9% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.25, a, 0.5\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma 0.25 a 0.5))

C

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

Julia

function code(a, b)
	return fma(0.25, a, 0.5)
end

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. inv-pow N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. prod-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-log.f64 98.8

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

4. Applied egg-rr 98.8%

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

5. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

      \[\leadsto e^{a + \color{blue}{\left(\mathsf{neg}\left(\log \left(1 + e^{a}\right)\right)\right)}} \]

   2. unsub-neg N/A

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

   3. exp-diff N/A

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

   4. remove-double-div N/A

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

   5. exp-neg N/A

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

   6. rem-exp-log N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. exp-neg N/A

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

   12. lft-mult-inverse N/A

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

   13. *-rgt-identity N/A

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

   14. lower-+.f64 N/A

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

   15. neg-mul-1 N/A

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

   16. lower-exp.f64 N/A

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

   17. neg-mul-1 N/A

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

   18. lower-neg.f64 62.9

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

7. Simplified 62.9%

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

8. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

   2. lower-fma.f64 39.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, a, 0.5\right)} \]

10. Simplified 39.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, a, 0.5\right)} \]
11. Add Preprocessing

Alternative 17: 39.7% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 84.2

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

5. Simplified 84.2%

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

6. Taylor expanded in b around 0

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

8. Simplified 39.3%

   \[\leadsto \color{blue}{0.5} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× 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 2024214 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ 1 (exp (- b a)))))

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