Harley's example

Percentage Accurate: 91.0% → 98.3%

Time: 1.3min

Alternatives: 10

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 2.00000000000000007e154
  2. c_n = 2
  3. t = 0.0
  4. s = 0.0

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: 91.0% 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: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s}\\ t_2 := e^{-t}\\ \mathbf{if}\;c\_p \leq 6 \cdot 10^{-71}:\\ \;\;\;\;e^{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + t\_1}\right) - \mathsf{log1p}\left(\frac{-1}{1 + t\_2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - t\_1}\right) - c\_p \cdot \mathsf{log1p}\left(t\_1\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - t\_2}\right)\right) + c\_p \cdot \mathsf{log1p}\left(t\_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- s))) (t_2 (exp (- t))))
   (if (<= c_p 6e-71)
     (exp
      (* c_n (- (log1p (/ -1.0 (+ 1.0 t_1))) (log1p (/ -1.0 (+ 1.0 t_2))))))
     (exp
      (+
       (-
        (- (* c_n (log1p (/ 1.0 (- -1.0 t_1)))) (* c_p (log1p t_1)))
        (* c_n (log1p (/ 1.0 (- -1.0 t_2)))))
       (* c_p (log1p t_2)))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-s);
	double t_2 = exp(-t);
	double tmp;
	if (c_p <= 6e-71) {
		tmp = exp((c_n * (log1p((-1.0 / (1.0 + t_1))) - log1p((-1.0 / (1.0 + t_2))))));
	} else {
		tmp = exp(((((c_n * log1p((1.0 / (-1.0 - t_1)))) - (c_p * log1p(t_1))) - (c_n * log1p((1.0 / (-1.0 - t_2))))) + (c_p * log1p(t_2))));
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = Math.exp(-s);
	double t_2 = Math.exp(-t);
	double tmp;
	if (c_p <= 6e-71) {
		tmp = Math.exp((c_n * (Math.log1p((-1.0 / (1.0 + t_1))) - Math.log1p((-1.0 / (1.0 + t_2))))));
	} else {
		tmp = Math.exp(((((c_n * Math.log1p((1.0 / (-1.0 - t_1)))) - (c_p * Math.log1p(t_1))) - (c_n * Math.log1p((1.0 / (-1.0 - t_2))))) + (c_p * Math.log1p(t_2))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = math.exp(-s)
	t_2 = math.exp(-t)
	tmp = 0
	if c_p <= 6e-71:
		tmp = math.exp((c_n * (math.log1p((-1.0 / (1.0 + t_1))) - math.log1p((-1.0 / (1.0 + t_2))))))
	else:
		tmp = math.exp(((((c_n * math.log1p((1.0 / (-1.0 - t_1)))) - (c_p * math.log1p(t_1))) - (c_n * math.log1p((1.0 / (-1.0 - t_2))))) + (c_p * math.log1p(t_2))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-s))
	t_2 = exp(Float64(-t))
	tmp = 0.0
	if (c_p <= 6e-71)
		tmp = exp(Float64(c_n * Float64(log1p(Float64(-1.0 / Float64(1.0 + t_1))) - log1p(Float64(-1.0 / Float64(1.0 + t_2))))));
	else
		tmp = exp(Float64(Float64(Float64(Float64(c_n * log1p(Float64(1.0 / Float64(-1.0 - t_1)))) - Float64(c_p * log1p(t_1))) - Float64(c_n * log1p(Float64(1.0 / Float64(-1.0 - t_2))))) + Float64(c_p * log1p(t_2))));
	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[Exp[(-t)], $MachinePrecision]}, If[LessEqual[c$95$p, 6e-71], N[Exp[N[(c$95$n * N[(N[Log[1 + N[(-1.0 / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(-1.0 / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(N[(N[(c$95$n * N[Log[1 + N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(c$95$p * N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c$95$n * N[Log[1 + N[(1.0 / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * N[Log[1 + t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 6.0000000000000003e-71

   1. Initial program 92.9%

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

      1. associate-/l/ 92.9%

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

   3. Simplified 92.9%

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

      1. add-exp-log 92.9%

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

      2. log-div 92.9%

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

   6. Applied egg-rr 95.7%

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

   7. Taylor expanded in c_n around inf 99.5%

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

      1. sub-neg 99.5%

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

      2. log1p-undefine 99.5%

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

      3. distribute-neg-frac 99.5%

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

      4. metadata-eval 99.5%

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

      5. sub-neg 99.5%

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

      6. log1p-define 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   9. Simplified 99.5%

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

   if 6.0000000000000003e-71 < c_p

   1. Initial program 81.1%

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

      1. associate-/l/ 81.1%

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

   3. Simplified 81.1%

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

      1. add-exp-log 81.1%

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

      2. log-div 81.1%

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

   6. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{e^{\left(\left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - e^{-s}}\right) + c\_p \cdot \left(-\mathsf{log1p}\left(e^{-s}\right)\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - e^{-t}}\right)\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 6 \cdot 10^{-71}:\\ \;\;\;\;e^{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - e^{-s}}\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - e^{-t}}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{-t}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 5 \cdot 10^{-165}:\\ \;\;\;\;e^{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot \left(0.5 + s \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 5e-165)
   (exp
    (*
     c_n
     (-
      (log1p (/ -1.0 (+ 1.0 (exp (- s)))))
      (log1p (/ -1.0 (+ 1.0 (exp (- t))))))))
   (pow
    (/ 1.0 (+ 2.0 (* s (+ -1.0 (* s (+ 0.5 (* s -0.16666666666666666)))))))
    c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 5e-165) {
		tmp = exp((c_n * (log1p((-1.0 / (1.0 + exp(-s)))) - log1p((-1.0 / (1.0 + exp(-t)))))));
	} else {
		tmp = pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p);
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 5e-165) {
		tmp = Math.exp((c_n * (Math.log1p((-1.0 / (1.0 + Math.exp(-s)))) - Math.log1p((-1.0 / (1.0 + Math.exp(-t)))))));
	} else {
		tmp = Math.pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 5e-165:
		tmp = math.exp((c_n * (math.log1p((-1.0 / (1.0 + math.exp(-s)))) - math.log1p((-1.0 / (1.0 + math.exp(-t)))))))
	else:
		tmp = math.pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 5e-165)
		tmp = exp(Float64(c_n * Float64(log1p(Float64(-1.0 / Float64(1.0 + exp(Float64(-s))))) - log1p(Float64(-1.0 / Float64(1.0 + exp(Float64(-t))))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(-1.0 + Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))))))) ^ c_p;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 5e-165], N[Exp[N[(c$95$n * N[(N[Log[1 + N[(-1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(-1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(-1.0 + N[(s * N[(0.5 + N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 4.99999999999999981e-165

   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. Step-by-step derivation

      1. associate-/l/ 90.7%

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

   3. Simplified 90.7%

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

      1. add-exp-log 90.7%

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

      2. log-div 90.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in c_n around inf 99.5%

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

      1. sub-neg 99.5%

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

      2. log1p-undefine 99.5%

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

      3. distribute-neg-frac 99.5%

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

      4. metadata-eval 99.5%

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

      5. sub-neg 99.5%

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

      6. log1p-define 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   9. Simplified 99.5%

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

   if 4.99999999999999981e-165 < (neg.f64 s)

   1. Initial program 85.7%

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

      1. associate-/l* 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in c_n around 0 87.3%

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

   6. Taylor expanded in c_p around 0 93.1%

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

   7. Taylor expanded in s around 0 96.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 5 \cdot 10^{-165}:\\ \;\;\;\;e^{c\_n \cdot \left(\mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot \left(0.5 + s \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+51}:\\ \;\;\;\;{\left(-1 + \left(1 + \frac{-1}{1 + e^{s}}\right)\right)}^{c\_p}\\ \mathbf{elif}\;-s \leq 5 \cdot 10^{-165}:\\ \;\;\;\;e^{c\_n \cdot \left(\log 0.5 - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot \left(0.5 + s \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) -2e+51)
   (pow (+ -1.0 (+ 1.0 (/ -1.0 (+ 1.0 (exp s))))) c_p)
   (if (<= (- s) 5e-165)
     (exp (* c_n (- (log 0.5) (log1p (/ -1.0 (+ 1.0 (exp (- t))))))))
     (pow
      (/ 1.0 (+ 2.0 (* s (+ -1.0 (* s (+ 0.5 (* s -0.16666666666666666)))))))
      c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= -2e+51) {
		tmp = pow((-1.0 + (1.0 + (-1.0 / (1.0 + exp(s))))), c_p);
	} else if (-s <= 5e-165) {
		tmp = exp((c_n * (log(0.5) - log1p((-1.0 / (1.0 + exp(-t)))))));
	} else {
		tmp = pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p);
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= -2e+51) {
		tmp = Math.pow((-1.0 + (1.0 + (-1.0 / (1.0 + Math.exp(s))))), c_p);
	} else if (-s <= 5e-165) {
		tmp = Math.exp((c_n * (Math.log(0.5) - Math.log1p((-1.0 / (1.0 + Math.exp(-t)))))));
	} else {
		tmp = Math.pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= -2e+51:
		tmp = math.pow((-1.0 + (1.0 + (-1.0 / (1.0 + math.exp(s))))), c_p)
	elif -s <= 5e-165:
		tmp = math.exp((c_n * (math.log(0.5) - math.log1p((-1.0 / (1.0 + math.exp(-t)))))))
	else:
		tmp = math.pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= -2e+51)
		tmp = Float64(-1.0 + Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(s))))) ^ c_p;
	elseif (Float64(-s) <= 5e-165)
		tmp = exp(Float64(c_n * Float64(log(0.5) - log1p(Float64(-1.0 / Float64(1.0 + exp(Float64(-t))))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(-1.0 + Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))))))) ^ c_p;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), -2e+51], N[Power[N[(-1.0 + N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], If[LessEqual[(-s), 5e-165], N[Exp[N[(c$95$n * N[(N[Log[0.5], $MachinePrecision] - N[Log[1 + N[(-1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(-1.0 + N[(s * N[(0.5 + N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (neg.f64 s) < -2e51

   1. Initial program 33.3%

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

      1. associate-/l* 33.3%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in c_n around 0 2.6%

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

   6. Taylor expanded in c_p around 0 3.1%

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

      1. expm1-log1p-u 3.1%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + e^{-s}}\right)\right)\right)}}^{c\_p}}{1} \]

      2. expm1-undefine 3.1%

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

   8. Applied egg-rr 100.0%

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

   if -2e51 < (neg.f64 s) < 4.99999999999999981e-165

   1. Initial program 92.6%

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

      1. associate-/l/ 92.6%

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

   3. Simplified 92.6%

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

      1. add-exp-log 92.6%

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

      2. log-div 92.6%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in c_n around inf 99.4%

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

      1. sub-neg 99.4%

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

      2. log1p-undefine 99.4%

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

      3. distribute-neg-frac 99.4%

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

      4. metadata-eval 99.4%

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

      5. sub-neg 99.4%

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

      6. log1p-define 99.4%

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

      7. distribute-neg-frac 99.4%

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

      8. metadata-eval 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in s around 0 99.4%

       \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 0.5 - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 99.4%

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

       2. sub-neg 99.4%

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

       3. log1p-define 99.4%

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

       4. distribute-neg-frac 99.4%

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

       5. metadata-eval 99.4%

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

   12. Simplified 99.4%

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

   if 4.99999999999999981e-165 < (neg.f64 s)

   1. Initial program 85.7%

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

      1. associate-/l* 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in c_n around 0 87.3%

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

   6. Taylor expanded in c_p around 0 93.1%

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

   7. Taylor expanded in s around 0 96.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+51}:\\ \;\;\;\;{\left(-1 + \left(1 + \frac{-1}{1 + e^{s}}\right)\right)}^{c\_p}\\ \mathbf{elif}\;-s \leq 5 \cdot 10^{-165}:\\ \;\;\;\;e^{c\_n \cdot \left(\log 0.5 - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot \left(0.5 + s \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+51}:\\ \;\;\;\;{\left(-1 + \left(1 + \frac{-1}{1 + e^{s}}\right)\right)}^{c\_p}\\ \mathbf{elif}\;-s \leq 5 \cdot 10^{-165}:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5 + t \cdot \left(-0.005208333333333333 \cdot \left(c\_n \cdot {t}^{2}\right) + c\_n \cdot 0.125\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot \left(0.5 + s \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) -2e+51)
   (pow (+ -1.0 (+ 1.0 (/ -1.0 (+ 1.0 (exp s))))) c_p)
   (if (<= (- s) 5e-165)
     (exp
      (*
       t
       (+
        (* c_n 0.5)
        (*
         t
         (+ (* -0.005208333333333333 (* c_n (pow t 2.0))) (* c_n 0.125))))))
     (pow
      (/ 1.0 (+ 2.0 (* s (+ -1.0 (* s (+ 0.5 (* s -0.16666666666666666)))))))
      c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= -2e+51) {
		tmp = pow((-1.0 + (1.0 + (-1.0 / (1.0 + exp(s))))), c_p);
	} else if (-s <= 5e-165) {
		tmp = exp((t * ((c_n * 0.5) + (t * ((-0.005208333333333333 * (c_n * pow(t, 2.0))) + (c_n * 0.125))))));
	} else {
		tmp = pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p);
	}
	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 <= (-2d+51)) then
        tmp = ((-1.0d0) + (1.0d0 + ((-1.0d0) / (1.0d0 + exp(s))))) ** c_p
    else if (-s <= 5d-165) then
        tmp = exp((t * ((c_n * 0.5d0) + (t * (((-0.005208333333333333d0) * (c_n * (t ** 2.0d0))) + (c_n * 0.125d0))))))
    else
        tmp = (1.0d0 / (2.0d0 + (s * ((-1.0d0) + (s * (0.5d0 + (s * (-0.16666666666666666d0)))))))) ** c_p
    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 <= -2e+51) {
		tmp = Math.pow((-1.0 + (1.0 + (-1.0 / (1.0 + Math.exp(s))))), c_p);
	} else if (-s <= 5e-165) {
		tmp = Math.exp((t * ((c_n * 0.5) + (t * ((-0.005208333333333333 * (c_n * Math.pow(t, 2.0))) + (c_n * 0.125))))));
	} else {
		tmp = Math.pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= -2e+51:
		tmp = math.pow((-1.0 + (1.0 + (-1.0 / (1.0 + math.exp(s))))), c_p)
	elif -s <= 5e-165:
		tmp = math.exp((t * ((c_n * 0.5) + (t * ((-0.005208333333333333 * (c_n * math.pow(t, 2.0))) + (c_n * 0.125))))))
	else:
		tmp = math.pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= -2e+51)
		tmp = Float64(-1.0 + Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(s))))) ^ c_p;
	elseif (Float64(-s) <= 5e-165)
		tmp = exp(Float64(t * Float64(Float64(c_n * 0.5) + Float64(t * Float64(Float64(-0.005208333333333333 * Float64(c_n * (t ^ 2.0))) + Float64(c_n * 0.125))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(-1.0 + Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))))))) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= -2e+51)
		tmp = (-1.0 + (1.0 + (-1.0 / (1.0 + exp(s))))) ^ c_p;
	elseif (-s <= 5e-165)
		tmp = exp((t * ((c_n * 0.5) + (t * ((-0.005208333333333333 * (c_n * (t ^ 2.0))) + (c_n * 0.125))))));
	else
		tmp = (1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), -2e+51], N[Power[N[(-1.0 + N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], If[LessEqual[(-s), 5e-165], N[Exp[N[(t * N[(N[(c$95$n * 0.5), $MachinePrecision] + N[(t * N[(N[(-0.005208333333333333 * N[(c$95$n * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c$95$n * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(-1.0 + N[(s * N[(0.5 + N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (neg.f64 s) < -2e51

   1. Initial program 33.3%

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

      1. associate-/l* 33.3%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in c_n around 0 2.6%

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

   6. Taylor expanded in c_p around 0 3.1%

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

      1. expm1-log1p-u 3.1%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + e^{-s}}\right)\right)\right)}}^{c\_p}}{1} \]

      2. expm1-undefine 3.1%

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

   8. Applied egg-rr 100.0%

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

   if -2e51 < (neg.f64 s) < 4.99999999999999981e-165

   1. Initial program 92.6%

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

      1. associate-/l/ 92.6%

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

   3. Simplified 92.6%

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

      1. add-exp-log 92.6%

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

      2. log-div 92.6%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in c_n around inf 99.4%

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

      1. sub-neg 99.4%

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

      2. log1p-undefine 99.4%

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

      3. distribute-neg-frac 99.4%

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

      4. metadata-eval 99.4%

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

      5. sub-neg 99.4%

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

      6. log1p-define 99.4%

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

      7. distribute-neg-frac 99.4%

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

      8. metadata-eval 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in s around 0 99.4%

       \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 0.5 - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 99.4%

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

       2. sub-neg 99.4%

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

       3. log1p-define 99.4%

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

       4. distribute-neg-frac 99.4%

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

       5. metadata-eval 99.4%

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

   12. Simplified 99.4%

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

   13. Taylor expanded in t around 0 99.2%

       \[\leadsto e^{\color{blue}{t \cdot \left(0.5 \cdot c\_n + t \cdot \left(-0.005208333333333333 \cdot \left(c\_n \cdot {t}^{2}\right) + 0.125 \cdot c\_n\right)\right)}} \]

   if 4.99999999999999981e-165 < (neg.f64 s)

   1. Initial program 85.7%

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

      1. associate-/l* 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in c_n around 0 87.3%

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

   6. Taylor expanded in c_p around 0 93.1%

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

   7. Taylor expanded in s around 0 96.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+51}:\\ \;\;\;\;{\left(-1 + \left(1 + \frac{-1}{1 + e^{s}}\right)\right)}^{c\_p}\\ \mathbf{elif}\;-s \leq 5 \cdot 10^{-165}:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5 + t \cdot \left(-0.005208333333333333 \cdot \left(c\_n \cdot {t}^{2}\right) + c\_n \cdot 0.125\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot \left(0.5 + s \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -1 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot \left(0.5 + s \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}\\ \mathbf{elif}\;s \leq 375:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \left(1 + \frac{-1}{1 + e^{s}}\right)\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -1e-161)
   (pow
    (/ 1.0 (+ 2.0 (* s (+ -1.0 (* s (+ 0.5 (* s -0.16666666666666666)))))))
    c_p)
   (if (<= s 375.0)
     (exp (* t (* c_n 0.5)))
     (pow (+ -1.0 (+ 1.0 (/ -1.0 (+ 1.0 (exp s))))) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -1e-161) {
		tmp = pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p);
	} else if (s <= 375.0) {
		tmp = exp((t * (c_n * 0.5)));
	} else {
		tmp = pow((-1.0 + (1.0 + (-1.0 / (1.0 + exp(s))))), c_p);
	}
	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 <= (-1d-161)) then
        tmp = (1.0d0 / (2.0d0 + (s * ((-1.0d0) + (s * (0.5d0 + (s * (-0.16666666666666666d0)))))))) ** c_p
    else if (s <= 375.0d0) then
        tmp = exp((t * (c_n * 0.5d0)))
    else
        tmp = ((-1.0d0) + (1.0d0 + ((-1.0d0) / (1.0d0 + exp(s))))) ** c_p
    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 <= -1e-161) {
		tmp = Math.pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p);
	} else if (s <= 375.0) {
		tmp = Math.exp((t * (c_n * 0.5)));
	} else {
		tmp = Math.pow((-1.0 + (1.0 + (-1.0 / (1.0 + Math.exp(s))))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -1e-161:
		tmp = math.pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p)
	elif s <= 375.0:
		tmp = math.exp((t * (c_n * 0.5)))
	else:
		tmp = math.pow((-1.0 + (1.0 + (-1.0 / (1.0 + math.exp(s))))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -1e-161)
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(-1.0 + Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))))))) ^ c_p;
	elseif (s <= 375.0)
		tmp = exp(Float64(t * Float64(c_n * 0.5)));
	else
		tmp = Float64(-1.0 + Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(s))))) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -1e-161)
		tmp = (1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))) ^ c_p;
	elseif (s <= 375.0)
		tmp = exp((t * (c_n * 0.5)));
	else
		tmp = (-1.0 + (1.0 + (-1.0 / (1.0 + exp(s))))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -1e-161], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(-1.0 + N[(s * N[(0.5 + N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], If[LessEqual[s, 375.0], N[Exp[N[(t * N[(c$95$n * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(-1.0 + N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if s < -1.00000000000000003e-161

   1. Initial program 85.5%

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

      1. associate-/l* 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in c_n around 0 87.1%

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

   6. Taylor expanded in c_p around 0 93.0%

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

   7. Taylor expanded in s around 0 96.2%

      \[\leadsto \frac{{\left(\frac{1}{\color{blue}{2 + s \cdot \left(s \cdot \left(0.5 + -0.16666666666666666 \cdot s\right) - 1\right)}}\right)}^{c\_p}}{1} \]

   if -1.00000000000000003e-161 < s < 375

   1. Initial program 93.6%

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

      1. associate-/l/ 93.6%

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

   3. Simplified 93.6%

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

      1. add-exp-log 93.6%

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

      2. log-div 93.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in c_n around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-undefine 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. metadata-eval 100.0%

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

      5. sub-neg 100.0%

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

      6. log1p-define 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in s around 0 100.0%

       \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 0.5 - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 100.0%

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

       2. sub-neg 100.0%

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

       3. log1p-define 100.0%

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

       4. distribute-neg-frac 100.0%

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

       5. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in t around 0 99.6%

       \[\leadsto e^{\color{blue}{0.5 \cdot \left(c\_n \cdot t\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 99.6%

          \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]

   15. Simplified 99.6%

       \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]

   if 375 < s

   1. Initial program 25.4%

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

      1. associate-/l* 25.4%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in c_n around 0 14.8%

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

   6. Taylor expanded in c_p around 0 15.2%

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

      1. expm1-log1p-u 15.2%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + e^{-s}}\right)\right)\right)}}^{c\_p}}{1} \]

      2. expm1-undefine 15.2%

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

   8. Applied egg-rr 87.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq -1 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot \left(0.5 + s \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}\\ \mathbf{elif}\;s \leq 375:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(-1 + \left(1 + \frac{-1}{1 + e^{s}}\right)\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -1 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot \left(0.5 + s \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -1e-161)
   (pow
    (/ 1.0 (+ 2.0 (* s (+ -1.0 (* s (+ 0.5 (* s -0.16666666666666666)))))))
    c_p)
   (exp (* t (* c_n 0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -1e-161) {
		tmp = pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p);
	} else {
		tmp = exp((t * (c_n * 0.5)));
	}
	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 <= (-1d-161)) then
        tmp = (1.0d0 / (2.0d0 + (s * ((-1.0d0) + (s * (0.5d0 + (s * (-0.16666666666666666d0)))))))) ** c_p
    else
        tmp = exp((t * (c_n * 0.5d0)))
    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 <= -1e-161) {
		tmp = Math.pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p);
	} else {
		tmp = Math.exp((t * (c_n * 0.5)));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -1e-161:
		tmp = math.pow((1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))), c_p)
	else:
		tmp = math.exp((t * (c_n * 0.5)))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -1e-161)
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(-1.0 + Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))))))) ^ c_p;
	else
		tmp = exp(Float64(t * Float64(c_n * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -1e-161)
		tmp = (1.0 / (2.0 + (s * (-1.0 + (s * (0.5 + (s * -0.16666666666666666))))))) ^ c_p;
	else
		tmp = exp((t * (c_n * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -1e-161], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(-1.0 + N[(s * N[(0.5 + N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[Exp[N[(t * N[(c$95$n * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -1.00000000000000003e-161

   1. Initial program 85.5%

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

      1. associate-/l* 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in c_n around 0 87.1%

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

   6. Taylor expanded in c_p around 0 93.0%

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

   7. Taylor expanded in s around 0 96.2%

      \[\leadsto \frac{{\left(\frac{1}{\color{blue}{2 + s \cdot \left(s \cdot \left(0.5 + -0.16666666666666666 \cdot s\right) - 1\right)}}\right)}^{c\_p}}{1} \]

   if -1.00000000000000003e-161 < s

   1. Initial program 90.8%

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

      1. associate-/l/ 90.8%

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

   3. Simplified 90.8%

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

      1. add-exp-log 90.8%

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

      2. log-div 90.8%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in c_n around inf 99.5%

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

      1. sub-neg 99.5%

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

      2. log1p-undefine 99.5%

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

      3. distribute-neg-frac 99.5%

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

      4. metadata-eval 99.5%

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

      5. sub-neg 99.5%

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

      6. log1p-define 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in s around 0 97.0%

       \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 0.5 - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 97.0%

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

       2. sub-neg 97.0%

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

       3. log1p-define 97.0%

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

       4. distribute-neg-frac 97.0%

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

       5. metadata-eval 97.0%

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

   12. Simplified 97.0%

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

   13. Taylor expanded in t around 0 96.6%

       \[\leadsto e^{\color{blue}{0.5 \cdot \left(c\_n \cdot t\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 96.6%

          \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]

   15. Simplified 96.6%

       \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq -1 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot \left(0.5 + s \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -1.55 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot 0.5\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -1.55e-161)
   (pow (/ 1.0 (+ 2.0 (* s (+ -1.0 (* s 0.5))))) c_p)
   (exp (* t (* c_n 0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -1.55e-161) {
		tmp = pow((1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))), c_p);
	} else {
		tmp = exp((t * (c_n * 0.5)));
	}
	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 <= (-1.55d-161)) then
        tmp = (1.0d0 / (2.0d0 + (s * ((-1.0d0) + (s * 0.5d0))))) ** c_p
    else
        tmp = exp((t * (c_n * 0.5d0)))
    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 <= -1.55e-161) {
		tmp = Math.pow((1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))), c_p);
	} else {
		tmp = Math.exp((t * (c_n * 0.5)));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -1.55e-161:
		tmp = math.pow((1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))), c_p)
	else:
		tmp = math.exp((t * (c_n * 0.5)))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -1.55e-161)
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(-1.0 + Float64(s * 0.5))))) ^ c_p;
	else
		tmp = exp(Float64(t * Float64(c_n * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -1.55e-161)
		tmp = (1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))) ^ c_p;
	else
		tmp = exp((t * (c_n * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -1.55e-161], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(-1.0 + N[(s * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[Exp[N[(t * N[(c$95$n * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -1.5499999999999999e-161

   1. Initial program 85.5%

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

      1. associate-/l* 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in c_n around 0 87.1%

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

   6. Taylor expanded in c_p around 0 93.0%

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

   7. Taylor expanded in s around 0 96.2%

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

   if -1.5499999999999999e-161 < s

   1. Initial program 90.8%

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

      1. associate-/l/ 90.8%

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

   3. Simplified 90.8%

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

      1. add-exp-log 90.8%

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

      2. log-div 90.8%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in c_n around inf 99.5%

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

      1. sub-neg 99.5%

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

      2. log1p-undefine 99.5%

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

      3. distribute-neg-frac 99.5%

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

      4. metadata-eval 99.5%

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

      5. sub-neg 99.5%

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

      6. log1p-define 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in s around 0 97.0%

       \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 0.5 - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 97.0%

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

       2. sub-neg 97.0%

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

       3. log1p-define 97.0%

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

       4. distribute-neg-frac 97.0%

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

       5. metadata-eval 97.0%

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

   12. Simplified 97.0%

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

   13. Taylor expanded in t around 0 96.6%

       \[\leadsto e^{\color{blue}{0.5 \cdot \left(c\_n \cdot t\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 96.6%

          \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]

   15. Simplified 96.6%

       \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq -1.55 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot 0.5\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -1 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{1}{2 - s}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -1e-161) (pow (/ 1.0 (- 2.0 s)) c_p) (exp (* t (* c_n 0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -1e-161) {
		tmp = pow((1.0 / (2.0 - s)), c_p);
	} else {
		tmp = exp((t * (c_n * 0.5)));
	}
	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 <= (-1d-161)) then
        tmp = (1.0d0 / (2.0d0 - s)) ** c_p
    else
        tmp = exp((t * (c_n * 0.5d0)))
    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 <= -1e-161) {
		tmp = Math.pow((1.0 / (2.0 - s)), c_p);
	} else {
		tmp = Math.exp((t * (c_n * 0.5)));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -1e-161:
		tmp = math.pow((1.0 / (2.0 - s)), c_p)
	else:
		tmp = math.exp((t * (c_n * 0.5)))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -1e-161)
		tmp = Float64(1.0 / Float64(2.0 - s)) ^ c_p;
	else
		tmp = exp(Float64(t * Float64(c_n * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -1e-161)
		tmp = (1.0 / (2.0 - s)) ^ c_p;
	else
		tmp = exp((t * (c_n * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -1e-161], N[Power[N[(1.0 / N[(2.0 - s), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[Exp[N[(t * N[(c$95$n * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -1.00000000000000003e-161

   1. Initial program 85.5%

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

      1. associate-/l* 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in c_n around 0 87.1%

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

   6. Taylor expanded in c_p around 0 93.0%

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

   7. Taylor expanded in s around 0 94.6%

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

      1. neg-mul-1 94.6%

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

      2. unsub-neg 94.6%

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

   9. Simplified 94.6%

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

   if -1.00000000000000003e-161 < s

   1. Initial program 90.8%

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

      1. associate-/l/ 90.8%

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

   3. Simplified 90.8%

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

      1. add-exp-log 90.8%

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

      2. log-div 90.8%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in c_n around inf 99.5%

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

      1. sub-neg 99.5%

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

      2. log1p-undefine 99.5%

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

      3. distribute-neg-frac 99.5%

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

      4. metadata-eval 99.5%

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

      5. sub-neg 99.5%

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

      6. log1p-define 99.5%

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

      7. distribute-neg-frac 99.5%

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

      8. metadata-eval 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in s around 0 97.0%

       \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 0.5 - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 97.0%

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

       2. sub-neg 97.0%

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

       3. log1p-define 97.0%

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

       4. distribute-neg-frac 97.0%

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

       5. metadata-eval 97.0%

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

   12. Simplified 97.0%

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

   13. Taylor expanded in t around 0 96.6%

       \[\leadsto e^{\color{blue}{0.5 \cdot \left(c\_n \cdot t\right)}} \]
   14. Step-by-step derivation

       1. associate-*r* 96.6%

          \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]

   15. Simplified 96.6%

       \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq -1 \cdot 10^{-161}:\\ \;\;\;\;{\left(\frac{1}{2 - s}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ e^{t \cdot \left(c\_n \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (exp (* t (* c_n 0.5))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((t * (c_n * 0.5)));
}

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 = exp((t * (c_n * 0.5d0)))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp((t * (c_n * 0.5)));
}

Python

def code(c_p, c_n, t, s):
	return math.exp((t * (c_n * 0.5)))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(t * Float64(c_n * 0.5)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp((t * (c_n * 0.5)));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(t * N[(c$95$n * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 89.5%

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

   1. associate-/l/ 89.5%

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

3. Simplified 89.5%

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

   1. add-exp-log 89.5%

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

   2. log-div 89.5%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in c_n around inf 96.6%

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

   1. sub-neg 96.6%

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

   2. log1p-undefine 96.6%

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

   3. distribute-neg-frac 96.6%

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

   4. metadata-eval 96.6%

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

   5. sub-neg 96.6%

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

   6. log1p-define 96.6%

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

   7. distribute-neg-frac 96.6%

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

   8. metadata-eval 96.6%

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

9. Simplified 96.6%

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

10. Taylor expanded in s around 0 94.7%

    \[\leadsto e^{\color{blue}{c\_n \cdot \left(\log 0.5 - \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)}} \]
11. Step-by-step derivation

    1. *-commutative 94.7%

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

    2. sub-neg 94.7%

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

    3. log1p-define 94.7%

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

    4. distribute-neg-frac 94.7%

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

    5. metadata-eval 94.7%

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

12. Simplified 94.7%

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

13. Taylor expanded in t around 0 94.4%

    \[\leadsto e^{\color{blue}{0.5 \cdot \left(c\_n \cdot t\right)}} \]
14. Step-by-step derivation

    1. associate-*r* 94.4%

       \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]

15. Simplified 94.4%

    \[\leadsto e^{\color{blue}{\left(0.5 \cdot c\_n\right) \cdot t}} \]

16. Final simplification 94.4%

    \[\leadsto e^{t \cdot \left(c\_n \cdot 0.5\right)} \]
17. Add Preprocessing

Alternative 10: 94.1% accurate, 835.0× 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 89.5%

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

   1. associate-/l* 89.5%

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

3. Simplified 89.5%

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

5. Taylor expanded in c_n around 0 92.4%

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

6. Taylor expanded in c_p around 0 93.2%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer target: 96.5% accurate, 1.3× 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 2024096 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

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