Harley's example

Percentage Accurate: 90.9% → 98.2%

Time: 55.6s

Alternatives: 9

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 3.744709066429704e-129
  2. c_n = 8.087723446513879e-153
  3. t = -8.701045206017167e-151
  4. s = -4.588353280742705e+179

Specification

\[0 < c\_p \land 0 < c\_n\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t))))) (t_2 (/ 1.0 (+ 1.0 (exp (- s))))))
   (/
    (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
    (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + exp(-t));
	double t_2 = 1.0 / (1.0 + exp(-s));
	return (pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    code = ((t_2 ** c_p) * ((1.0d0 - t_2) ** c_n)) / ((t_1 ** c_p) * ((1.0d0 - t_1) ** c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + Math.exp(-t));
	double t_2 = 1.0 / (1.0 + Math.exp(-s));
	return (Math.pow(t_2, c_p) * Math.pow((1.0 - t_2), c_n)) / (Math.pow(t_1, c_p) * Math.pow((1.0 - t_1), c_n));
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 / (1.0 + math.exp(-t))
	t_2 = 1.0 / (1.0 + math.exp(-s))
	return (math.pow(t_2, c_p) * math.pow((1.0 - t_2), c_n)) / (math.pow(t_1, c_p) * math.pow((1.0 - t_1), c_n))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	return Float64(Float64((t_2 ^ c_p) * (Float64(1.0 - t_2) ^ c_n)) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = ((t_2 ^ c_p) * ((1.0 - t_2) ^ c_n)) / ((t_1 ^ c_p) * ((1.0 - t_1) ^ c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \frac{{t\_2}^{c\_p} \cdot {\left(1 - t\_2\right)}^{c\_n}}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t))))) (t_2 (/ 1.0 (+ 1.0 (exp (- s))))))
   (/
    (* (pow t_2 c_p) (pow (- 1.0 t_2) c_n))
    (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + exp(-t));
	double t_2 = 1.0 / (1.0 + exp(-s));
	return (pow(t_2, c_p) * pow((1.0 - t_2), c_n)) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    real(8) :: t_2
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    code = ((t_2 ** c_p) * ((1.0d0 - t_2) ** c_n)) / ((t_1 ** c_p) * ((1.0d0 - t_1) ** c_n))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (1.0 + Math.exp(-t));
	double t_2 = 1.0 / (1.0 + Math.exp(-s));
	return (Math.pow(t_2, c_p) * Math.pow((1.0 - t_2), c_n)) / (Math.pow(t_1, c_p) * Math.pow((1.0 - t_1), c_n));
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 / (1.0 + math.exp(-t))
	t_2 = 1.0 / (1.0 + math.exp(-s))
	return (math.pow(t_2, c_p) * math.pow((1.0 - t_2), c_n)) / (math.pow(t_1, c_p) * math.pow((1.0 - t_1), c_n))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	return Float64(Float64((t_2 ^ c_p) * (Float64(1.0 - t_2) ^ c_n)) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = ((t_2 ^ c_p) * ((1.0 - t_2) ^ c_n)) / ((t_1 ^ c_p) * ((1.0 - t_1) ^ c_n));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 98.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s} + 1\\ \mathbf{if}\;-s \leq -2 \cdot 10^{-212}:\\ \;\;\;\;{\left(e^{c\_n}\right)}^{\left(\mathsf{log1p}\left(\frac{-1}{t\_1}\right) - \log 0.5\right)}\\ \mathbf{elif}\;-s \leq 20000000:\\ \;\;\;\;e^{\mathsf{fma}\left(\log 0.5, c\_n, \left(-c\_n\right) \cdot \mathsf{log1p}\left(\frac{-1}{e^{-t} + 1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{t\_1}\right)}^{c\_p}}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (+ (exp (- s)) 1.0)))
   (if (<= (- s) -2e-212)
     (pow (exp c_n) (- (log1p (/ -1.0 t_1)) (log 0.5)))
     (if (<= (- s) 20000000.0)
       (exp
        (fma (log 0.5) c_n (* (- c_n) (log1p (/ -1.0 (+ (exp (- t)) 1.0))))))
       (/ (pow (/ 1.0 t_1) c_p) 1.0)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-s) + 1.0;
	double tmp;
	if (-s <= -2e-212) {
		tmp = pow(exp(c_n), (log1p((-1.0 / t_1)) - log(0.5)));
	} else if (-s <= 20000000.0) {
		tmp = exp(fma(log(0.5), c_n, (-c_n * log1p((-1.0 / (exp(-t) + 1.0))))));
	} else {
		tmp = pow((1.0 / t_1), c_p) / 1.0;
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(exp(Float64(-s)) + 1.0)
	tmp = 0.0
	if (Float64(-s) <= -2e-212)
		tmp = exp(c_n) ^ Float64(log1p(Float64(-1.0 / t_1)) - log(0.5));
	elseif (Float64(-s) <= 20000000.0)
		tmp = exp(fma(log(0.5), c_n, Float64(Float64(-c_n) * log1p(Float64(-1.0 / Float64(exp(Float64(-t)) + 1.0))))));
	else
		tmp = Float64((Float64(1.0 / t_1) ^ c_p) / 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[(-s), -2e-212], N[Power[N[Exp[c$95$n], $MachinePrecision], N[(N[Log[1 + N[(-1.0 / t$95$1), $MachinePrecision]], $MachinePrecision] - N[Log[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[(-s), 20000000.0], N[Exp[N[(N[Log[0.5], $MachinePrecision] * c$95$n + N[((-c$95$n) * N[Log[1 + N[(-1.0 / N[(N[Exp[(-t)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[(1.0 / t$95$1), $MachinePrecision], c$95$p], $MachinePrecision] / 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (neg.f64 s) < -1.99999999999999991e-212

   1. Initial program 87.2%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \color{blue}{\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. +-commutative N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      9. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 90.1%

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + t \cdot \left(\frac{1}{48} \cdot {t}^{2} - \frac{1}{4}\right)\right)}^{c\_n}} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.0%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}^{c\_n}} \]

   12. Taylor expanded in t around 0

       \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{\color{blue}{{\frac{1}{2}}^{c\_n}}} \]
   13. Step-by-step derivation

   14. Applied rewrites 98.8%

       \[\leadsto {\left(e^{c\_n}\right)}^{\color{blue}{\left(\mathsf{log1p}\left(\frac{-1}{e^{-s} + 1}\right) - \log 0.5\right)}} \]

   if -1.99999999999999991e-212 < (neg.f64 s) < 2e7

   1. Initial program 92.6%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \color{blue}{\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. +-commutative N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      9. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.1%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]

   10. Applied rewrites 99.6%

       \[\leadsto e^{\mathsf{fma}\left(\log 0.5, c\_n, -\mathsf{log1p}\left(\frac{-1}{e^{-t} + 1}\right) \cdot c\_n\right)} \]

   if 2e7 < (neg.f64 s)

   1. Initial program 66.7%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. +-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}} \]

      12. +-commutative N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)}} + 1}\right)}^{c\_p}} \]

      15. lower-neg.f64 83.3

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{\color{blue}{-t}} + 1}\right)}^{c\_p}} \]

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}}} \]

   6. Taylor expanded in c_p around 0

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{-212}:\\ \;\;\;\;{\left(e^{c\_n}\right)}^{\left(\mathsf{log1p}\left(\frac{-1}{e^{-s} + 1}\right) - \log 0.5\right)}\\ \mathbf{elif}\;-s \leq 20000000:\\ \;\;\;\;e^{\mathsf{fma}\left(\log 0.5, c\_n, \left(-c\_n\right) \cdot \mathsf{log1p}\left(\frac{-1}{e^{-t} + 1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 20000000:\\ \;\;\;\;e^{\mathsf{fma}\left(\log 0.5, c\_n, \left(-c\_n\right) \cdot \mathsf{log1p}\left(\frac{-1}{e^{-t} + 1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 20000000.0)
   (exp (fma (log 0.5) c_n (* (- c_n) (log1p (/ -1.0 (+ (exp (- t)) 1.0))))))
   (/ (pow (/ 1.0 (+ (exp (- s)) 1.0)) c_p) 1.0)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 20000000.0) {
		tmp = exp(fma(log(0.5), c_n, (-c_n * log1p((-1.0 / (exp(-t) + 1.0))))));
	} else {
		tmp = pow((1.0 / (exp(-s) + 1.0)), c_p) / 1.0;
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 20000000.0)
		tmp = exp(fma(log(0.5), c_n, Float64(Float64(-c_n) * log1p(Float64(-1.0 / Float64(exp(Float64(-t)) + 1.0))))));
	else
		tmp = Float64((Float64(1.0 / Float64(exp(Float64(-s)) + 1.0)) ^ c_p) / 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 20000000.0], N[Exp[N[(N[Log[0.5], $MachinePrecision] * c$95$n + N[((-c$95$n) * N[Log[1 + N[(-1.0 / N[(N[Exp[(-t)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[(1.0 / N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 2e7

   1. Initial program 90.6%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \color{blue}{\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. +-commutative N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      9. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.5%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]

   10. Applied rewrites 97.4%

       \[\leadsto e^{\mathsf{fma}\left(\log 0.5, c\_n, -\mathsf{log1p}\left(\frac{-1}{e^{-t} + 1}\right) \cdot c\_n\right)} \]

   if 2e7 < (neg.f64 s)

   1. Initial program 66.7%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. +-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}} \]

      12. +-commutative N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)}} + 1}\right)}^{c\_p}} \]

      15. lower-neg.f64 83.3

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{\color{blue}{-t}} + 1}\right)}^{c\_p}} \]

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}}} \]

   6. Taylor expanded in c_p around 0

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 20000000:\\ \;\;\;\;e^{\mathsf{fma}\left(\log 0.5, c\_n, \left(-c\_n\right) \cdot \mathsf{log1p}\left(\frac{-1}{e^{-t} + 1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{e^{-t} + 1}\\ \mathbf{if}\;-t \leq 5 \cdot 10^{-107}:\\ \;\;\;\;\frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{{t\_1}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 - t\_1\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ (exp (- t)) 1.0))))
   (if (<= (- t) 5e-107)
     (/ (pow (/ 1.0 (fma (fma 0.5 s -1.0) s 2.0)) c_p) (pow t_1 c_p))
     (/ (pow 0.5 c_n) (pow (- 1.0 t_1) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 / (exp(-t) + 1.0);
	double tmp;
	if (-t <= 5e-107) {
		tmp = pow((1.0 / fma(fma(0.5, s, -1.0), s, 2.0)), c_p) / pow(t_1, c_p);
	} else {
		tmp = pow(0.5, c_n) / pow((1.0 - t_1), c_n);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(exp(Float64(-t)) + 1.0))
	tmp = 0.0
	if (Float64(-t) <= 5e-107)
		tmp = Float64((Float64(1.0 / fma(fma(0.5, s, -1.0), s, 2.0)) ^ c_p) / (t_1 ^ c_p));
	else
		tmp = Float64((0.5 ^ c_n) / (Float64(1.0 - t_1) ^ c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 / N[(N[Exp[(-t)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[(-t), 5e-107], N[(N[Power[N[(1.0 / N[(N[(0.5 * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] / N[Power[t$95$1, c$95$p], $MachinePrecision]), $MachinePrecision], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 4.99999999999999971e-107

   1. Initial program 92.4%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. +-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}} \]

      12. +-commutative N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)}} + 1}\right)}^{c\_p}} \]

      15. lower-neg.f64 94.7

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{\color{blue}{-t}} + 1}\right)}^{c\_p}} \]

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.4%

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]

   if 4.99999999999999971e-107 < (neg.f64 t)

   1. Initial program 81.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \color{blue}{\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. +-commutative N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      9. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(t\_1 \cdot t\_1\right)}^{\left(\left(-0.5\right) \cdot c\_p\right)}}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (fma (fma 0.5 s -1.0) s 2.0)))
   (if (<= t -1e-68)
     (/ (pow 0.5 c_n) (pow (- 1.0 (/ 1.0 (+ (exp (- t)) 1.0))) c_n))
     (/ (pow (* t_1 t_1) (* (- 0.5) c_p)) 1.0))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = fma(fma(0.5, s, -1.0), s, 2.0);
	double tmp;
	if (t <= -1e-68) {
		tmp = pow(0.5, c_n) / pow((1.0 - (1.0 / (exp(-t) + 1.0))), c_n);
	} else {
		tmp = pow((t_1 * t_1), (-0.5 * c_p)) / 1.0;
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = fma(fma(0.5, s, -1.0), s, 2.0)
	tmp = 0.0
	if (t <= -1e-68)
		tmp = Float64((0.5 ^ c_n) / (Float64(1.0 - Float64(1.0 / Float64(exp(Float64(-t)) + 1.0))) ^ c_n));
	else
		tmp = Float64((Float64(t_1 * t_1) ^ Float64(Float64(-0.5) * c_p)) / 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[(0.5 * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision]}, If[LessEqual[t, -1e-68], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(1.0 - N[(1.0 / N[(N[Exp[(-t)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$1 * t$95$1), $MachinePrecision], N[((-0.5) * c$95$p), $MachinePrecision]], $MachinePrecision] / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.00000000000000007e-68

   1. Initial program 70.1%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \color{blue}{\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. +-commutative N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      9. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]

   if -1.00000000000000007e-68 < t

   1. Initial program 93.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. +-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}} \]

      12. +-commutative N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)}} + 1}\right)}^{c\_p}} \]

      15. lower-neg.f64 95.2

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{\color{blue}{-t}} + 1}\right)}^{c\_p}} \]

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.7%

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]

   9. Taylor expanded in c_p around 0

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, s, -1\right), s, 2\right)}\right)}^{c\_p}}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 95.4%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 96.3%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(-1 \cdot \left(c\_p \cdot 0.5\right)\right)}}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(\left(-0.5\right) \cdot c\_p\right)}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(t\_1 \cdot t\_1\right)}^{\left(\left(-0.5\right) \cdot c\_p\right)}}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (fma (fma 0.5 s -1.0) s 2.0)))
   (if (<= t -1e-68)
     (/ (pow 0.5 c_n) (pow (fma -0.25 t 0.5) c_n))
     (/ (pow (* t_1 t_1) (* (- 0.5) c_p)) 1.0))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = fma(fma(0.5, s, -1.0), s, 2.0);
	double tmp;
	if (t <= -1e-68) {
		tmp = pow(0.5, c_n) / pow(fma(-0.25, t, 0.5), c_n);
	} else {
		tmp = pow((t_1 * t_1), (-0.5 * c_p)) / 1.0;
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = fma(fma(0.5, s, -1.0), s, 2.0)
	tmp = 0.0
	if (t <= -1e-68)
		tmp = Float64((0.5 ^ c_n) / (fma(-0.25, t, 0.5) ^ c_n));
	else
		tmp = Float64((Float64(t_1 * t_1) ^ Float64(Float64(-0.5) * c_p)) / 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[(0.5 * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision]}, If[LessEqual[t, -1e-68], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(-0.25 * t + 0.5), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$1 * t$95$1), $MachinePrecision], N[((-0.5) * c$95$p), $MachinePrecision]], $MachinePrecision] / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.00000000000000007e-68

   1. Initial program 70.1%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \color{blue}{\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. +-commutative N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      9. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}} \]

   if -1.00000000000000007e-68 < t

   1. Initial program 93.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. +-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}} \]

      12. +-commutative N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)}} + 1}\right)}^{c\_p}} \]

      15. lower-neg.f64 95.2

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{\color{blue}{-t}} + 1}\right)}^{c\_p}} \]

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.7%

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]

   9. Taylor expanded in c_p around 0

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, s, -1\right), s, 2\right)}\right)}^{c\_p}}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 95.4%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 96.3%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(-1 \cdot \left(c\_p \cdot 0.5\right)\right)}}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-68}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(\left(-0.5\right) \cdot c\_p\right)}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\\ \mathbf{if}\;c\_n \leq 2000000000:\\ \;\;\;\;\frac{{\left(t\_1 \cdot t\_1\right)}^{\left(\left(-0.5\right) \cdot c\_p\right)}}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (fma (fma 0.5 s -1.0) s 2.0)))
   (if (<= c_n 2000000000.0)
     (/ (pow (* t_1 t_1) (* (- 0.5) c_p)) 1.0)
     (/ 1.0 (pow (fma (fma (* t t) 0.020833333333333332 -0.25) t 0.5) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = fma(fma(0.5, s, -1.0), s, 2.0);
	double tmp;
	if (c_n <= 2000000000.0) {
		tmp = pow((t_1 * t_1), (-0.5 * c_p)) / 1.0;
	} else {
		tmp = 1.0 / pow(fma(fma((t * t), 0.020833333333333332, -0.25), t, 0.5), c_n);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = fma(fma(0.5, s, -1.0), s, 2.0)
	tmp = 0.0
	if (c_n <= 2000000000.0)
		tmp = Float64((Float64(t_1 * t_1) ^ Float64(Float64(-0.5) * c_p)) / 1.0);
	else
		tmp = Float64(1.0 / (fma(fma(Float64(t * t), 0.020833333333333332, -0.25), t, 0.5) ^ c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[(0.5 * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision]}, If[LessEqual[c$95$n, 2000000000.0], N[(N[Power[N[(t$95$1 * t$95$1), $MachinePrecision], N[((-0.5) * c$95$p), $MachinePrecision]], $MachinePrecision] / 1.0), $MachinePrecision], N[(1.0 / N[Power[N[(N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] * t + 0.5), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c_n < 2e9

   1. Initial program 91.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. +-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}} \]

      12. +-commutative N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)}} + 1}\right)}^{c\_p}} \]

      15. lower-neg.f64 94.0

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{\color{blue}{-t}} + 1}\right)}^{c\_p}} \]

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.4%

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]

   9. Taylor expanded in c_p around 0

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, s, -1\right), s, 2\right)}\right)}^{c\_p}}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 95.5%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 96.2%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(-1 \cdot \left(c\_p \cdot 0.5\right)\right)}}{1} \]

   if 2e9 < c_n

   1. Initial program 16.7%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \color{blue}{\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. +-commutative N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      9. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + t \cdot \left(\frac{1}{48} \cdot {t}^{2} - \frac{1}{4}\right)\right)}^{c\_n}} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}^{c\_n}} \]

   12. Taylor expanded in c_n around 0

       \[\leadsto \frac{1}{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{-1}{4}\right), t, \frac{1}{2}\right)\right)}}^{c\_n}} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto \frac{1}{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}}^{c\_n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_n \leq 2000000000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(\left(-0.5\right) \cdot c\_p\right)}}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_n \leq 2000000000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_n 2000000000.0)
   (/ (pow (fma (fma 0.5 s -1.0) s 2.0) (- c_p)) 1.0)
   (/ 1.0 (pow (fma (fma (* t t) 0.020833333333333332 -0.25) t 0.5) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_n <= 2000000000.0) {
		tmp = pow(fma(fma(0.5, s, -1.0), s, 2.0), -c_p) / 1.0;
	} else {
		tmp = 1.0 / pow(fma(fma((t * t), 0.020833333333333332, -0.25), t, 0.5), c_n);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_n <= 2000000000.0)
		tmp = Float64((fma(fma(0.5, s, -1.0), s, 2.0) ^ Float64(-c_p)) / 1.0);
	else
		tmp = Float64(1.0 / (fma(fma(Float64(t * t), 0.020833333333333332, -0.25), t, 0.5) ^ c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$n, 2000000000.0], N[(N[Power[N[(N[(0.5 * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / 1.0), $MachinePrecision], N[(1.0 / N[Power[N[(N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] * t + 0.5), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_n < 2e9

   1. Initial program 91.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. +-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}} \]

      12. +-commutative N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)}} + 1}\right)}^{c\_p}} \]

      15. lower-neg.f64 94.0

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{\color{blue}{-t}} + 1}\right)}^{c\_p}} \]

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.4%

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]

   9. Taylor expanded in c_p around 0

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, s, -1\right), s, 2\right)}\right)}^{c\_p}}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 95.5%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 95.5%

       \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1}} \]

   if 2e9 < c_n

   1. Initial program 16.7%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \color{blue}{\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. +-commutative N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      9. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + t \cdot \left(\frac{1}{48} \cdot {t}^{2} - \frac{1}{4}\right)\right)}^{c\_n}} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}^{c\_n}} \]

   12. Taylor expanded in c_n around 0

       \[\leadsto \frac{1}{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{-1}{4}\right), t, \frac{1}{2}\right)\right)}}^{c\_n}} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto \frac{1}{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}}^{c\_n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_n \leq 2000000000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_n \leq 2 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_n 2e+15)
   1.0
   (/ 1.0 (pow (fma (fma (* t t) 0.020833333333333332 -0.25) t 0.5) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_n <= 2e+15) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / pow(fma(fma((t * t), 0.020833333333333332, -0.25), t, 0.5), c_n);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_n <= 2e+15)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / (fma(fma(Float64(t * t), 0.020833333333333332, -0.25), t, 0.5) ^ c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$n, 2e+15], 1.0, N[(1.0 / N[Power[N[(N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + -0.25), $MachinePrecision] * t + 0.5), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_n < 2e15

   1. Initial program 91.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. +-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      8. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      10. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}} \]

      12. +-commutative N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

      14. lower-exp.f64 N/A

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)}} + 1}\right)}^{c\_p}} \]

      15. lower-neg.f64 94.0

          \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{\color{blue}{-t}} + 1}\right)}^{c\_p}} \]

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}}} \]

   6. Taylor expanded in c_p around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 95.2%

      \[\leadsto 1 \]

   if 2e15 < c_n

   1. Initial program 16.7%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \color{blue}{\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. +-commutative N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      9. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      10. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{e^{-s} + 1}\right)}^{c\_n}}{{\left(1 - \frac{1}{e^{-t} + 1}\right)}^{c\_n}}} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\color{blue}{1} - \frac{1}{e^{-t} + 1}\right)}^{c\_n}} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + t \cdot \left(\frac{1}{48} \cdot {t}^{2} - \frac{1}{4}\right)\right)}^{c\_n}} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}^{c\_n}} \]

   12. Taylor expanded in c_n around 0

       \[\leadsto \frac{1}{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{-1}{4}\right), t, \frac{1}{2}\right)\right)}}^{c\_n}} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto \frac{1}{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, -0.25\right), t, 0.5\right)\right)}}^{c\_n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 93.9% accurate, 896.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 1.0)

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = 1.0d0
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0;
}

Python

def code(c_p, c_n, t, s):
	return 1.0

Julia

function code(c_p, c_n, t, s)
	return 1.0
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := 1.0

Derivation

1. Initial program 90.1%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. Add Preprocessing

3. Taylor expanded in c_n around 0

   \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   4. +-commutative N/A

      \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(s\right)} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   5. neg-mul-1 N/A

      \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s} + 1}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   7. lower-exp.f64 N/A

      \[\leadsto \frac{{\left(\frac{1}{\color{blue}{e^{-1 \cdot s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   8. neg-mul-1 N/A

      \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{\mathsf{neg}\left(s\right)}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   9. lower-neg.f64 N/A

      \[\leadsto \frac{{\left(\frac{1}{e^{\color{blue}{-s}} + 1}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{\color{blue}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}} \]

   12. +-commutative N/A

       \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

   13. lower-+.f64 N/A

       \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)} + 1}}\right)}^{c\_p}} \]

   14. lower-exp.f64 N/A

       \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{e^{\mathsf{neg}\left(t\right)}} + 1}\right)}^{c\_p}} \]

   15. lower-neg.f64 91.8

       \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{\color{blue}{-t}} + 1}\right)}^{c\_p}} \]

5. Applied rewrites 91.8%

   \[\leadsto \color{blue}{\frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}}} \]

6. Taylor expanded in c_p around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 93.1%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 96.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1 + e^{-t}}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(\frac{1 + e^{t}}{1 + e^{s}}\right)}^{c\_n} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (*
  (pow (/ (+ 1.0 (exp (- t))) (+ 1.0 (exp (- s)))) c_p)
  (pow (/ (+ 1.0 (exp t)) (+ 1.0 (exp s))) c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	return pow(((1.0 + exp(-t)) / (1.0 + exp(-s))), c_p) * pow(((1.0 + exp(t)) / (1.0 + exp(s))), c_n);
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = (((1.0d0 + exp(-t)) / (1.0d0 + exp(-s))) ** c_p) * (((1.0d0 + exp(t)) / (1.0d0 + exp(s))) ** c_n)
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow(((1.0 + Math.exp(-t)) / (1.0 + Math.exp(-s))), c_p) * Math.pow(((1.0 + Math.exp(t)) / (1.0 + Math.exp(s))), c_n);
}

Python

def code(c_p, c_n, t, s):
	return math.pow(((1.0 + math.exp(-t)) / (1.0 + math.exp(-s))), c_p) * math.pow(((1.0 + math.exp(t)) / (1.0 + math.exp(s))), c_n)

Julia

function code(c_p, c_n, t, s)
	return Float64((Float64(Float64(1.0 + exp(Float64(-t))) / Float64(1.0 + exp(Float64(-s)))) ^ c_p) * (Float64(Float64(1.0 + exp(t)) / Float64(1.0 + exp(s))) ^ c_n))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = (((1.0 + exp(-t)) / (1.0 + exp(-s))) ^ c_p) * (((1.0 + exp(t)) / (1.0 + exp(s))) ^ c_n);
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[Power[N[(N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] * N[Power[N[(N[(1.0 + N[Exp[t], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024276 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

  :alt
  (! :herbie-platform default (* (pow (/ (+ 1 (exp (- t))) (+ 1 (exp (- s)))) c_p) (pow (/ (+ 1 (exp t)) (+ 1 (exp s))) c_n)))

  (/ (* (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- s))))) c_n)) (* (pow (/ 1.0 (+ 1.0 (exp (- t)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- t))))) c_n))))