Harley's example

Percentage Accurate: 90.6% → 98.0%

Time: 53.6s

Alternatives: 10

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 1.2645730441944144e-80
  2. c_n = 3.455145643175537e-283
  3. t = 4.988760692266297e+66
  4. s = -2.3954921145740213e-141

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: 90.6% 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: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s}\\ t_2 := \frac{1}{1 + t\_1}\\ t_3 := e^{-t}\\ \mathbf{if}\;-t \leq 5 \cdot 10^{-11}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(t\_3\right) - \mathsf{log1p}\left(t\_1\right), \left(\mathsf{log1p}\left({\left(-1 - t\_1\right)}^{-1}\right) - \mathsf{log1p}\left({\left(-1 - t\_3\right)}^{-1}\right)\right) \cdot c\_n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 - t\_2\right)}^{c\_n} \cdot {t\_2}^{c\_p}}{{\left(1 - \frac{1}{1 + t\_3}\right)}^{c\_n} \cdot \left({0.5}^{c\_p} \cdot \mathsf{fma}\left(t, 0.5 \cdot c\_p, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- s))) (t_2 (/ 1.0 (+ 1.0 t_1))) (t_3 (exp (- t))))
   (if (<= (- t) 5e-11)
     (exp
      (fma
       c_p
       (- (log1p t_3) (log1p t_1))
       (*
        (- (log1p (pow (- -1.0 t_1) -1.0)) (log1p (pow (- -1.0 t_3) -1.0)))
        c_n)))
     (/
      (* (pow (- 1.0 t_2) c_n) (pow t_2 c_p))
      (*
       (pow (- 1.0 (/ 1.0 (+ 1.0 t_3))) c_n)
       (* (pow 0.5 c_p) (fma t (* 0.5 c_p) 1.0)))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-s);
	double t_2 = 1.0 / (1.0 + t_1);
	double t_3 = exp(-t);
	double tmp;
	if (-t <= 5e-11) {
		tmp = exp(fma(c_p, (log1p(t_3) - log1p(t_1)), ((log1p(pow((-1.0 - t_1), -1.0)) - log1p(pow((-1.0 - t_3), -1.0))) * c_n)));
	} else {
		tmp = (pow((1.0 - t_2), c_n) * pow(t_2, c_p)) / (pow((1.0 - (1.0 / (1.0 + t_3))), c_n) * (pow(0.5, c_p) * fma(t, (0.5 * c_p), 1.0)));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-s))
	t_2 = Float64(1.0 / Float64(1.0 + t_1))
	t_3 = exp(Float64(-t))
	tmp = 0.0
	if (Float64(-t) <= 5e-11)
		tmp = exp(fma(c_p, Float64(log1p(t_3) - log1p(t_1)), Float64(Float64(log1p((Float64(-1.0 - t_1) ^ -1.0)) - log1p((Float64(-1.0 - t_3) ^ -1.0))) * c_n)));
	else
		tmp = Float64(Float64((Float64(1.0 - t_2) ^ c_n) * (t_2 ^ c_p)) / Float64((Float64(1.0 - Float64(1.0 / Float64(1.0 + t_3))) ^ c_n) * Float64((0.5 ^ c_p) * fma(t, 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[Exp[(-s)], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Exp[(-t)], $MachinePrecision]}, If[LessEqual[(-t), 5e-11], N[Exp[N[(c$95$p * N[(N[Log[1 + t$95$3], $MachinePrecision] - N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[1 + N[Power[N[(-1.0 - t$95$1), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[Power[N[(-1.0 - t$95$3), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c$95$n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision] * N[Power[t$95$2, c$95$p], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(1.0 - N[(1.0 / N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] * N[(N[Power[0.5, c$95$p], $MachinePrecision] * N[(t * N[(0.5 * c$95$p), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 5.00000000000000018e-11

   1. Initial program 94.6%

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

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \left(\mathsf{log1p}\left({\left(-1 - e^{-s}\right)}^{-1}\right) - \mathsf{log1p}\left({\left(-1 - e^{-t}\right)}^{-1}\right)\right)\right)}} \]

   if 5.00000000000000018e-11 < (neg.f64 t)

   1. Initial program 36.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(\frac{1}{2} \cdot \left(c\_p \cdot \color{blue}{\left({\frac{1}{2}}^{c\_p} \cdot t\right)}\right) + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(\frac{1}{2} \cdot \color{blue}{\left(\left(c\_p \cdot {\frac{1}{2}}^{c\_p}\right) \cdot t\right)} + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(\color{blue}{\left(\frac{1}{2} \cdot \left(c\_p \cdot {\frac{1}{2}}^{c\_p}\right)\right) \cdot t} + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      4. *-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(\color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(c\_p \cdot {\frac{1}{2}}^{c\_p}\right)\right)} + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(t \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot c\_p\right) \cdot {\frac{1}{2}}^{c\_p}\right)} + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(\color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot c\_p\right)\right) \cdot {\frac{1}{2}}^{c\_p}} + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      7. distribute-lft1-in N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\color{blue}{\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_p\right) + 1\right) \cdot {\frac{1}{2}}^{c\_p}\right)} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\color{blue}{\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_p\right) + 1\right) \cdot {\frac{1}{2}}^{c\_p}\right)} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 5 \cdot 10^{-11}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right), \left(\mathsf{log1p}\left({\left(-1 - e^{-s}\right)}^{-1}\right) - \mathsf{log1p}\left({\left(-1 - e^{-t}\right)}^{-1}\right)\right) \cdot c\_n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n} \cdot {\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n} \cdot \left({0.5}^{c\_p} \cdot \mathsf{fma}\left(t, 0.5 \cdot c\_p, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := {\left(1 - t\_1\right)}^{c\_n}\\ t_3 := \frac{1}{1 + e^{-s}}\\ t_4 := {\left(1 - t\_3\right)}^{c\_n} \cdot {t\_3}^{c\_p}\\ \mathbf{if}\;\frac{t\_4}{{t\_1}^{c\_p} \cdot t\_2} \leq 2:\\ \;\;\;\;\frac{t\_4}{t\_2 \cdot \left({0.5}^{c\_p} \cdot \mathsf{fma}\left(t, 0.5 \cdot c\_p, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t)))))
        (t_2 (pow (- 1.0 t_1) c_n))
        (t_3 (/ 1.0 (+ 1.0 (exp (- s)))))
        (t_4 (* (pow (- 1.0 t_3) c_n) (pow t_3 c_p))))
   (if (<= (/ t_4 (* (pow t_1 c_p) t_2)) 2.0)
     (/ t_4 (* t_2 (* (pow 0.5 c_p) (fma t (* 0.5 c_p) 1.0))))
     (/
      (pow (fma (fma (fma -0.16666666666666666 s 0.5) s -1.0) s 2.0) (- c_p))
      1.0))))

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 = pow((1.0 - t_1), c_n);
	double t_3 = 1.0 / (1.0 + exp(-s));
	double t_4 = pow((1.0 - t_3), c_n) * pow(t_3, c_p);
	double tmp;
	if ((t_4 / (pow(t_1, c_p) * t_2)) <= 2.0) {
		tmp = t_4 / (t_2 * (pow(0.5, c_p) * fma(t, (0.5 * c_p), 1.0)));
	} else {
		tmp = pow(fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0), -c_p) / 1.0;
	}
	return tmp;
}

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 - t_1) ^ c_n
	t_3 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	t_4 = Float64((Float64(1.0 - t_3) ^ c_n) * (t_3 ^ c_p))
	tmp = 0.0
	if (Float64(t_4 / Float64((t_1 ^ c_p) * t_2)) <= 2.0)
		tmp = Float64(t_4 / Float64(t_2 * Float64((0.5 ^ c_p) * fma(t, Float64(0.5 * c_p), 1.0))));
	else
		tmp = Float64((fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0) ^ Float64(-c_p)) / 1.0);
	end
	return tmp
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[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(1.0 - t$95$3), $MachinePrecision], c$95$n], $MachinePrecision] * N[Power[t$95$3, c$95$p], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$4 / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$4 / N[(t$95$2 * N[(N[Power[0.5, c$95$p], $MachinePrecision] * N[(t * N[(0.5 * c$95$p), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(-0.16666666666666666 * s + 0.5), $MachinePrecision] * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n))) < 2

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(\frac{1}{2} \cdot \left(c\_p \cdot \color{blue}{\left({\frac{1}{2}}^{c\_p} \cdot t\right)}\right) + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(\frac{1}{2} \cdot \color{blue}{\left(\left(c\_p \cdot {\frac{1}{2}}^{c\_p}\right) \cdot t\right)} + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(\color{blue}{\left(\frac{1}{2} \cdot \left(c\_p \cdot {\frac{1}{2}}^{c\_p}\right)\right) \cdot t} + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      4. *-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(\color{blue}{t \cdot \left(\frac{1}{2} \cdot \left(c\_p \cdot {\frac{1}{2}}^{c\_p}\right)\right)} + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(t \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot c\_p\right) \cdot {\frac{1}{2}}^{c\_p}\right)} + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\left(\color{blue}{\left(t \cdot \left(\frac{1}{2} \cdot c\_p\right)\right) \cdot {\frac{1}{2}}^{c\_p}} + {\frac{1}{2}}^{c\_p}\right) \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      7. distribute-lft1-in N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\color{blue}{\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_p\right) + 1\right) \cdot {\frac{1}{2}}^{c\_p}\right)} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{\color{blue}{\left(\left(t \cdot \left(\frac{1}{2} \cdot c\_p\right) + 1\right) \cdot {\frac{1}{2}}^{c\_p}\right)} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 2 < (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n)))

   1. Initial program 0.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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 31.7

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

   5. Applied rewrites 31.7%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 66.3%

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

   10. Applied rewrites 66.3%

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

   11. Taylor expanded in s around 0

       \[\leadsto \frac{{\left(2 + s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 83.0%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n} \cdot {\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \leq 2:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n} \cdot {\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n} \cdot \left({0.5}^{c\_p} \cdot \mathsf{fma}\left(t, 0.5 \cdot c\_p, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ t_3 := {t\_2}^{c\_p}\\ t_4 := {\left(1 - t\_2\right)}^{c\_n}\\ \mathbf{if}\;\frac{t\_4 \cdot t\_3}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \leq 2:\\ \;\;\;\;t\_4 \cdot \frac{t\_3}{{0.5}^{c\_n} \cdot {0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \end{array} \]

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)))))
        (t_3 (pow t_2 c_p))
        (t_4 (pow (- 1.0 t_2) c_n)))
   (if (<= (/ (* t_4 t_3) (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n))) 2.0)
     (* t_4 (/ t_3 (* (pow 0.5 c_n) (pow 0.5 c_p))))
     (/
      (pow (fma (fma (fma -0.16666666666666666 s 0.5) s -1.0) s 2.0) (- c_p))
      1.0))))

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));
	double t_3 = pow(t_2, c_p);
	double t_4 = pow((1.0 - t_2), c_n);
	double tmp;
	if (((t_4 * t_3) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n))) <= 2.0) {
		tmp = t_4 * (t_3 / (pow(0.5, c_n) * pow(0.5, c_p)));
	} else {
		tmp = pow(fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0), -c_p) / 1.0;
	}
	return tmp;
}

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))))
	t_3 = t_2 ^ c_p
	t_4 = Float64(1.0 - t_2) ^ c_n
	tmp = 0.0
	if (Float64(Float64(t_4 * t_3) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n))) <= 2.0)
		tmp = Float64(t_4 * Float64(t_3 / Float64((0.5 ^ c_n) * (0.5 ^ c_p))));
	else
		tmp = Float64((fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0) ^ Float64(-c_p)) / 1.0);
	end
	return tmp
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]}, Block[{t$95$3 = N[Power[t$95$2, c$95$p], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$4 * t$95$3), $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], 2.0], N[(t$95$4 * N[(t$95$3 / N[(N[Power[0.5, c$95$n], $MachinePrecision] * N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(-0.16666666666666666 * s + 0.5), $MachinePrecision] * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n))) < 2

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n} \cdot \frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\frac{1}{2}}^{c\_n} \cdot {\frac{1}{2}}^{c\_p}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\frac{1}{2}}^{c\_n} \cdot {\frac{1}{2}}^{c\_p}} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\frac{1}{2}}^{c\_n} \cdot {\frac{1}{2}}^{c\_p}} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}} \]

   5. Applied rewrites 99.5%

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

   if 2 < (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n)))

   1. Initial program 0.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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 31.7

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

   5. Applied rewrites 31.7%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 66.3%

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

   10. Applied rewrites 66.3%

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

   11. Taylor expanded in s around 0

       \[\leadsto \frac{{\left(2 + s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 83.0%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n} \cdot {\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \leq 2:\\ \;\;\;\;{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n} \cdot \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{0.5}^{c\_n} \cdot {0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ t_3 := {t\_2}^{c\_p}\\ \mathbf{if}\;\frac{{\left(1 - t\_2\right)}^{c\_n} \cdot t\_3}{{t\_1}^{c\_p} \cdot {\left(1 - t\_1\right)}^{c\_n}} \leq 1:\\ \;\;\;\;\frac{t\_3}{\mathsf{fma}\left(\left({0.5}^{c\_p} \cdot t\right) \cdot c\_p, 0.5, {0.5}^{c\_p}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \end{array} \]

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)))))
        (t_3 (pow t_2 c_p)))
   (if (<=
        (/
         (* (pow (- 1.0 t_2) c_n) t_3)
         (* (pow t_1 c_p) (pow (- 1.0 t_1) c_n)))
        1.0)
     (/ t_3 (fma (* (* (pow 0.5 c_p) t) c_p) 0.5 (pow 0.5 c_p)))
     (/
      (pow (fma (fma (fma -0.16666666666666666 s 0.5) s -1.0) s 2.0) (- c_p))
      1.0))))

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));
	double t_3 = pow(t_2, c_p);
	double tmp;
	if (((pow((1.0 - t_2), c_n) * t_3) / (pow(t_1, c_p) * pow((1.0 - t_1), c_n))) <= 1.0) {
		tmp = t_3 / fma(((pow(0.5, c_p) * t) * c_p), 0.5, pow(0.5, c_p));
	} else {
		tmp = pow(fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0), -c_p) / 1.0;
	}
	return tmp;
}

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))))
	t_3 = t_2 ^ c_p
	tmp = 0.0
	if (Float64(Float64((Float64(1.0 - t_2) ^ c_n) * t_3) / Float64((t_1 ^ c_p) * (Float64(1.0 - t_1) ^ c_n))) <= 1.0)
		tmp = Float64(t_3 / fma(Float64(Float64((0.5 ^ c_p) * t) * c_p), 0.5, (0.5 ^ c_p)));
	else
		tmp = Float64((fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0) ^ Float64(-c_p)) / 1.0);
	end
	return tmp
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]}, Block[{t$95$3 = N[Power[t$95$2, c$95$p], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(1.0 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision] * t$95$3), $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], 1.0], N[(t$95$3 / N[(N[(N[(N[Power[0.5, c$95$p], $MachinePrecision] * t), $MachinePrecision] * c$95$p), $MachinePrecision] * 0.5 + N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(-0.16666666666666666 * s + 0.5), $MachinePrecision] * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n))) < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in c_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 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 99.6%

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

   if 1 < (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n)))

   1. Initial program 8.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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 34.2

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

   5. Applied rewrites 34.2%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 66.1%

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

   10. Applied rewrites 66.1%

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

   11. Taylor expanded in s around 0

       \[\leadsto \frac{{\left(2 + s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.5%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n} \cdot {\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \leq 1:\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{\mathsf{fma}\left(\left({0.5}^{c\_p} \cdot t\right) \cdot c\_p, 0.5, {0.5}^{c\_p}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := {t\_1}^{c\_p}\\ t_3 := \frac{1}{1 + e^{-s}}\\ t_4 := {t\_3}^{c\_p}\\ \mathbf{if}\;\frac{{\left(1 - t\_3\right)}^{c\_n} \cdot t\_4}{t\_2 \cdot {\left(1 - t\_1\right)}^{c\_n}} \leq 1:\\ \;\;\;\;\frac{t\_4}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (/ 1.0 (+ 1.0 (exp (- t)))))
        (t_2 (pow t_1 c_p))
        (t_3 (/ 1.0 (+ 1.0 (exp (- s)))))
        (t_4 (pow t_3 c_p)))
   (if (<= (/ (* (pow (- 1.0 t_3) c_n) t_4) (* t_2 (pow (- 1.0 t_1) c_n))) 1.0)
     (/ t_4 t_2)
     (/
      (pow (fma (fma (fma -0.16666666666666666 s 0.5) s -1.0) s 2.0) (- c_p))
      1.0))))

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 = pow(t_1, c_p);
	double t_3 = 1.0 / (1.0 + exp(-s));
	double t_4 = pow(t_3, c_p);
	double tmp;
	if (((pow((1.0 - t_3), c_n) * t_4) / (t_2 * pow((1.0 - t_1), c_n))) <= 1.0) {
		tmp = t_4 / t_2;
	} else {
		tmp = pow(fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0), -c_p) / 1.0;
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = t_1 ^ c_p
	t_3 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	t_4 = t_3 ^ c_p
	tmp = 0.0
	if (Float64(Float64((Float64(1.0 - t_3) ^ c_n) * t_4) / Float64(t_2 * (Float64(1.0 - t_1) ^ c_n))) <= 1.0)
		tmp = Float64(t_4 / t_2);
	else
		tmp = Float64((fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0) ^ Float64(-c_p)) / 1.0);
	end
	return tmp
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[Power[t$95$1, c$95$p], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, c$95$p], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(1.0 - t$95$3), $MachinePrecision], c$95$n], $MachinePrecision] * t$95$4), $MachinePrecision] / N[(t$95$2 * N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(t$95$4 / t$95$2), $MachinePrecision], N[(N[Power[N[(N[(N[(-0.16666666666666666 * s + 0.5), $MachinePrecision] * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n))) < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in c_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 99.6

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

   5. Applied rewrites 99.6%

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

   if 1 < (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n)))

   1. Initial program 8.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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 34.2

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

   5. Applied rewrites 34.2%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 66.1%

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

   10. Applied rewrites 66.1%

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

   11. Taylor expanded in s around 0

       \[\leadsto \frac{{\left(2 + s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.5%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n} \cdot {\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \leq 1:\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + e^{-s}\\ \mathbf{if}\;-s \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{t\_1}\right)}^{c\_n}}{1}\\ \mathbf{elif}\;-s \leq 10000000:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(s \cdot c\_p\right) \cdot c\_p\right) \cdot 0.125, s, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_1}^{\left(-c\_p\right)}}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (+ 1.0 (exp (- s)))))
   (if (<= (- s) -4e+17)
     (/ (pow (- 1.0 (/ 1.0 t_1)) c_n) 1.0)
     (if (<= (- s) 10000000.0)
       (fma (* (* (* s c_p) c_p) 0.125) s 1.0)
       (/ (pow t_1 (- c_p)) 1.0)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 + exp(-s);
	double tmp;
	if (-s <= -4e+17) {
		tmp = pow((1.0 - (1.0 / t_1)), c_n) / 1.0;
	} else if (-s <= 10000000.0) {
		tmp = fma((((s * c_p) * c_p) * 0.125), s, 1.0);
	} else {
		tmp = pow(t_1, -c_p) / 1.0;
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 + exp(Float64(-s)))
	tmp = 0.0
	if (Float64(-s) <= -4e+17)
		tmp = Float64((Float64(1.0 - Float64(1.0 / t_1)) ^ c_n) / 1.0);
	elseif (Float64(-s) <= 10000000.0)
		tmp = fma(Float64(Float64(Float64(s * c_p) * c_p) * 0.125), s, 1.0);
	else
		tmp = Float64((t_1 ^ Float64(-c_p)) / 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[(-s), -4e+17], N[(N[Power[N[(1.0 - N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] / 1.0), $MachinePrecision], If[LessEqual[(-s), 10000000.0], N[(N[(N[(N[(s * c$95$p), $MachinePrecision] * c$95$p), $MachinePrecision] * 0.125), $MachinePrecision] * s + 1.0), $MachinePrecision], N[(N[Power[t$95$1, (-c$95$p)], $MachinePrecision] / 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (neg.f64 s) < -4e17

   1. Initial program 20.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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 60.0%

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

   6. Taylor expanded in c_n around 0

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

   8. Applied rewrites 100.0%

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

   if -4e17 < (neg.f64 s) < 1e7

   1. Initial program 93.1%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n} \cdot \frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\frac{1}{2}}^{c\_n} \cdot {\frac{1}{2}}^{c\_p}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\frac{1}{2}}^{c\_n} \cdot {\frac{1}{2}}^{c\_p}} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\frac{1}{2}}^{c\_n} \cdot {\frac{1}{2}}^{c\_p}} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}} \]

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in s around 0

      \[\leadsto 1 + \color{blue}{s \cdot \left(\frac{-1}{2} \cdot c\_n + \left(\frac{1}{2} \cdot c\_p + s \cdot \left(\frac{-1}{4} \cdot \left(c\_n \cdot c\_p\right) + \left(\frac{-1}{8} \cdot c\_n + \left(\frac{-1}{8} \cdot c\_p + \left(\frac{1}{8} \cdot {c\_n}^{2} + \frac{1}{8} \cdot {c\_p}^{2}\right)\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot c\_p, c\_n, \mathsf{fma}\left(0.125, \mathsf{fma}\left(c\_p, c\_p, c\_n \cdot c\_n\right), -0.125 \cdot \left(c\_p + c\_n\right)\right)\right), s, \mathsf{fma}\left(-0.5, c\_n, c\_p \cdot 0.5\right)\right), \color{blue}{s}, 1\right) \]

   9. Taylor expanded in c_p around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{8} \cdot \left({c\_p}^{2} \cdot s\right), s, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 98.1%

       \[\leadsto \mathsf{fma}\left(\left(\left(s \cdot c\_p\right) \cdot c\_p\right) \cdot 0.125, s, 1\right) \]

   if 1e7 < (neg.f64 s)

   1. Initial program 62.5%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   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%

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

   10. Applied rewrites 100.0%

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

   12. Applied rewrites 100.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -4 \cdot 10^{+17}:\\ \;\;\;\;\frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1}\\ \mathbf{elif}\;-s \leq 10000000:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(s \cdot c\_p\right) \cdot c\_p\right) \cdot 0.125, s, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + e^{-s}\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 10000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + e^{-s}\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 10000000.0) 1.0 (/ (pow (+ 1.0 (exp (- s))) (- c_p)) 1.0)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 10000000.0) {
		tmp = 1.0;
	} else {
		tmp = pow((1.0 + exp(-s)), -c_p) / 1.0;
	}
	return tmp;
}

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) :: tmp
    if (-s <= 10000000.0d0) then
        tmp = 1.0d0
    else
        tmp = ((1.0d0 + exp(-s)) ** -c_p) / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 10000000.0) {
		tmp = 1.0;
	} else {
		tmp = Math.pow((1.0 + Math.exp(-s)), -c_p) / 1.0;
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 10000000.0:
		tmp = 1.0
	else:
		tmp = math.pow((1.0 + math.exp(-s)), -c_p) / 1.0
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 10000000.0)
		tmp = 1.0;
	else
		tmp = Float64((Float64(1.0 + exp(Float64(-s))) ^ Float64(-c_p)) / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 10000000.0)
		tmp = 1.0;
	else
		tmp = ((1.0 + exp(-s)) ^ -c_p) / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 10000000.0], 1.0, N[(N[Power[N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision], (-c$95$p)], $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 1e7

   1. Initial program 91.6%

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

   3. Taylor expanded in c_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 94.2

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 96.2%

      \[\leadsto 1 \]

   if 1e7 < (neg.f64 s)

   1. Initial program 62.5%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   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%

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

   10. Applied rewrites 100.0%

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

   12. Applied rewrites 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 10000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + e^{-s}\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot c\_p, c\_n, \mathsf{fma}\left(0.125, \mathsf{fma}\left(c\_p, c\_p, c\_n \cdot c\_n\right), \left(c\_n + c\_p\right) \cdot -0.125\right)\right), s, \mathsf{fma}\left(-0.5, c\_n, 0.5 \cdot c\_p\right)\right), s, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(s, 0.5, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 0.005)
   (fma
    (fma
     (fma
      (* -0.25 c_p)
      c_n
      (fma 0.125 (fma c_p c_p (* c_n c_n)) (* (+ c_n c_p) -0.125)))
     s
     (fma -0.5 c_n (* 0.5 c_p)))
    s
    1.0)
   (/ (pow (fma (fma s 0.5 -1.0) s 2.0) (- c_p)) 1.0)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 0.005) {
		tmp = fma(fma(fma((-0.25 * c_p), c_n, fma(0.125, fma(c_p, c_p, (c_n * c_n)), ((c_n + c_p) * -0.125))), s, fma(-0.5, c_n, (0.5 * c_p))), s, 1.0);
	} else {
		tmp = pow(fma(fma(s, 0.5, -1.0), s, 2.0), -c_p) / 1.0;
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 0.005)
		tmp = fma(fma(fma(Float64(-0.25 * c_p), c_n, fma(0.125, fma(c_p, c_p, Float64(c_n * c_n)), Float64(Float64(c_n + c_p) * -0.125))), s, fma(-0.5, c_n, Float64(0.5 * c_p))), s, 1.0);
	else
		tmp = Float64((fma(fma(s, 0.5, -1.0), s, 2.0) ^ Float64(-c_p)) / 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 0.005], N[(N[(N[(N[(-0.25 * c$95$p), $MachinePrecision] * c$95$n + N[(0.125 * N[(c$95$p * c$95$p + N[(c$95$n * c$95$n), $MachinePrecision]), $MachinePrecision] + N[(N[(c$95$n + c$95$p), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * s + N[(-0.5 * c$95$n + N[(0.5 * c$95$p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * s + 1.0), $MachinePrecision], N[(N[Power[N[(N[(s * 0.5 + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 0.0050000000000000001

   1. Initial program 92.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

         \[\leadsto \color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n} \cdot \frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\frac{1}{2}}^{c\_n} \cdot {\frac{1}{2}}^{c\_p}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\frac{1}{2}}^{c\_n} \cdot {\frac{1}{2}}^{c\_p}} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}}{{\frac{1}{2}}^{c\_n} \cdot {\frac{1}{2}}^{c\_p}} \cdot {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}} \]

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in s around 0

      \[\leadsto 1 + \color{blue}{s \cdot \left(\frac{-1}{2} \cdot c\_n + \left(\frac{1}{2} \cdot c\_p + s \cdot \left(\frac{-1}{4} \cdot \left(c\_n \cdot c\_p\right) + \left(\frac{-1}{8} \cdot c\_n + \left(\frac{-1}{8} \cdot c\_p + \left(\frac{1}{8} \cdot {c\_n}^{2} + \frac{1}{8} \cdot {c\_p}^{2}\right)\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot c\_p, c\_n, \mathsf{fma}\left(0.125, \mathsf{fma}\left(c\_p, c\_p, c\_n \cdot c\_n\right), -0.125 \cdot \left(c\_p + c\_n\right)\right)\right), s, \mathsf{fma}\left(-0.5, c\_n, c\_p \cdot 0.5\right)\right), \color{blue}{s}, 1\right) \]

   if 0.0050000000000000001 < (neg.f64 s)

   1. Initial program 54.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 88.1%

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

   10. Applied rewrites 88.1%

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

   11. Taylor expanded in s around 0

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

   13. Applied rewrites 88.2%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot c\_p, c\_n, \mathsf{fma}\left(0.125, \mathsf{fma}\left(c\_p, c\_p, c\_n \cdot c\_n\right), \left(c\_n + c\_p\right) \cdot -0.125\right)\right), s, \mathsf{fma}\left(-0.5, c\_n, 0.5 \cdot c\_p\right)\right), s, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(s, 0.5, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (/
  (pow (fma (fma (fma -0.16666666666666666 s 0.5) s -1.0) s 2.0) (- c_p))
  1.0))

C

double code(double c_p, double c_n, double t, double s) {
	return pow(fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0), -c_p) / 1.0;
}

Julia

function code(c_p, c_n, t, s)
	return Float64((fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0) ^ Float64(-c_p)) / 1.0)
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[Power[N[(N[(N[(-0.16666666666666666 * s + 0.5), $MachinePrecision] * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / 1.0), $MachinePrecision]

Derivation

1. Initial program 90.7%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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 93.2

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

5. Applied rewrites 93.2%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 93.2%

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

10. Applied rewrites 93.2%

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

11. Taylor expanded in s around 0

    \[\leadsto \frac{{\left(2 + s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1} \]
12. Step-by-step derivation

13. Applied rewrites 94.7%

    \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-1 \cdot c\_p\right)}}{1} \]

14. Final simplification 94.7%

    \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{1} \]
15. Add Preprocessing

Alternative 10: 93.8% 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 90.7%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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 93.2

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

5. Applied rewrites 93.2%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 93.3%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 96.4% 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 2024241 
(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))))