Harley's example

Percentage Accurate: 90.4% → 96.0%

Time: 57.6s

Alternatives: 11

Speedup: 64.0×

• could not determine a ground truth

  1. c_p = 4.5435737241847525e+75
  2. c_n = 1.6335090758022787e-144
  3. t = -9.551763935041827e-269
  4. s = 1.4959249919454316e+292

Specification

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

Math

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

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (let* ((t_1 (/ 1 (+ 1 (exp (- t))))) (t_2 (/ 1 (+ 1 (exp (- s))))))
  (/
   (* (pow t_2 c_p) (pow (- 1 t_2) c_n))
   (* (pow t_1 c_p) (pow (- 1 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    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 / N[(1 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1 / N[(1 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 90.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (let* ((t_1 (/ 1 (+ 1 (exp (- t))))) (t_2 (/ 1 (+ 1 (exp (- s))))))
  (/
   (* (pow t_2 c_p) (pow (- 1 t_2) c_n))
   (* (pow t_1 c_p) (pow (- 1 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    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 / N[(1 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1 / N[(1 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[t$95$2, c$95$p], $MachinePrecision] * N[Power[N[(1 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, c$95$p], $MachinePrecision] * N[Power[N[(1 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 96.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \mathbf{if}\;s \leq -3700000000000000158015604916224:\\ \;\;\;\;\frac{{t\_2}^{c\_p}}{1 + c\_p \cdot \log t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 - t\_2\right)}^{c\_n}}{{\left(1 - t\_1\right)}^{c\_n}}\\ \end{array} \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (let* ((t_1 (/ 1 (+ 1 (exp (- t))))) (t_2 (/ 1 (+ 1 (exp (- s))))))
  (if (<= s -3700000000000000158015604916224)
    (/ (pow t_2 c_p) (+ 1 (* c_p (log t_1))))
    (/ (pow (- 1 t_2) c_n) (pow (- 1 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));
	double tmp;
	if (s <= -3.7e+30) {
		tmp = pow(t_2, c_p) / (1.0 + (c_p * log(t_1)));
	} else {
		tmp = pow((1.0 - t_2), c_n) / pow((1.0 - t_1), c_n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    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
    real(8) :: tmp
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    if (s <= (-3.7d+30)) then
        tmp = (t_2 ** c_p) / (1.0d0 + (c_p * log(t_1)))
    else
        tmp = ((1.0d0 - t_2) ** c_n) / ((1.0d0 - t_1) ** c_n)
    end if
    code = tmp
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));
	double tmp;
	if (s <= -3.7e+30) {
		tmp = Math.pow(t_2, c_p) / (1.0 + (c_p * Math.log(t_1)));
	} else {
		tmp = Math.pow((1.0 - t_2), c_n) / Math.pow((1.0 - t_1), c_n);
	}
	return tmp;
}

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))
	tmp = 0
	if s <= -3.7e+30:
		tmp = math.pow(t_2, c_p) / (1.0 + (c_p * math.log(t_1)))
	else:
		tmp = math.pow((1.0 - t_2), c_n) / math.pow((1.0 - t_1), c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	tmp = 0.0
	if (s <= -3.7e+30)
		tmp = Float64((t_2 ^ c_p) / Float64(1.0 + Float64(c_p * log(t_1))));
	else
		tmp = Float64((Float64(1.0 - t_2) ^ c_n) / (Float64(1.0 - t_1) ^ c_n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = 0.0;
	if (s <= -3.7e+30)
		tmp = (t_2 ^ c_p) / (1.0 + (c_p * log(t_1)));
	else
		tmp = ((1.0 - t_2) ^ c_n) / ((1.0 - t_1) ^ c_n);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1 / N[(1 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1 / N[(1 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s, -3700000000000000158015604916224], N[(N[Power[t$95$2, c$95$p], $MachinePrecision] / N[(1 + N[(c$95$p * N[Log[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[N[(1 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if s < -3.7000000000000002e30

   1. Initial program 90.4%

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

   2. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in c_p around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 93.2%

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

   7. Applied rewrites 93.2%

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

   if -3.7000000000000002e30 < s

   1. Initial program 90.4%

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

   2. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

   4. Applied rewrites 94.2%

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

Alternative 2: 95.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_1 := \frac{1}{1 + e^{-s}}\\ \mathbf{if}\;s \leq -3700000000000000158015604916224:\\ \;\;\;\;\frac{{t\_1}^{c\_p}}{{\frac{1}{2}}^{c\_p}}\\ \mathbf{elif}\;s \leq \frac{35681192317649}{713623846352979940529142984724747568191373312}:\\ \;\;\;\;\frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 - t\_1\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\frac{1}{2}\right)}\\ \end{array} \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (let* ((t_1 (/ 1 (+ 1 (exp (- s))))))
  (if (<= s -3700000000000000158015604916224)
    (/ (pow t_1 c_p) (pow 1/2 c_p))
    (if (<=
         s
         35681192317649/713623846352979940529142984724747568191373312)
      (/ (pow 1/2 c_n) (pow (- 1 (/ 1 (+ 1 (exp (- t))))) c_n))
      (/ (pow (- 1 t_1) c_n) (+ 1 (* c_n (30-log1z0 1/2))))))))

Derivation

1. Split input into 3 regimes

2. if s < -3.7000000000000002e30

   1. Initial program 90.4%

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

   2. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 93.2%

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

   if -3.7000000000000002e30 < s < 5.0000000000000004e-32

   1. Initial program 90.4%

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

   2. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in s around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-neg.f64 94.0%

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

   7. Applied rewrites 94.0%

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

   if 5.0000000000000004e-32 < s

   1. Initial program 90.4%

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

   2. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in c_n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-30-log1z0 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      7. lower-neg.f64 93.3%

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)} \]

   7. Applied rewrites 93.3%

      \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + \color{blue}{c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\frac{1}{2}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.7%

       \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\frac{1}{2}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{1}{1 + e^{-s}}\\ \mathbf{if}\;s \leq -3700000000000000158015604916224:\\ \;\;\;\;\frac{{t\_1}^{c\_p}}{{\frac{1}{2}}^{c\_p}}\\ \mathbf{elif}\;s \leq \frac{35681192317649}{713623846352979940529142984724747568191373312}:\\ \;\;\;\;\frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 - t\_1\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\frac{1}{2}\right)}\\ \end{array} \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (let* ((t_1 (/ 1 (+ 1 (exp (- s))))))
  (if (<= s -3700000000000000158015604916224)
    (/ (pow t_1 c_p) (pow 1/2 c_p))
    (if (<=
         s
         35681192317649/713623846352979940529142984724747568191373312)
      (/ (pow 1/2 c_n) (pow (+ 1/2 (* -1/4 t)) c_n))
      (/ (pow (- 1 t_1) c_n) (+ 1 (* c_n (30-log1z0 1/2))))))))

Derivation

1. Split input into 3 regimes

2. if s < -3.7000000000000002e30

   1. Initial program 90.4%

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

   2. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 93.2%

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

   if -3.7000000000000002e30 < s < 5.0000000000000004e-32

   1. Initial program 90.4%

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

   2. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in s around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-neg.f64 94.0%

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]

      2. lower-*.f64 93.7%

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]

   10. Applied rewrites 93.7%

       \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]

   if 5.0000000000000004e-32 < s

   1. Initial program 90.4%

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

   2. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in c_n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-30-log1z0 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      7. lower-neg.f64 93.3%

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)} \]

   7. Applied rewrites 93.3%

      \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + \color{blue}{c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\frac{1}{2}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.7%

       \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\frac{1}{2}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := \frac{1}{1 + e^{-s}}\\ \mathbf{if}\;s \leq -3700000000000000158015604916224:\\ \;\;\;\;\frac{{t\_2}^{c\_p}}{{t\_1}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 - t\_2\right)}^{c\_n}}{{\left(1 - t\_1\right)}^{c\_n}}\\ \end{array} \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (let* ((t_1 (/ 1 (+ 1 (exp (- t))))) (t_2 (/ 1 (+ 1 (exp (- s))))))
  (if (<= s -3700000000000000158015604916224)
    (/ (pow t_2 c_p) (pow t_1 c_p))
    (/ (pow (- 1 t_2) c_n) (pow (- 1 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));
	double tmp;
	if (s <= -3.7e+30) {
		tmp = pow(t_2, c_p) / pow(t_1, c_p);
	} else {
		tmp = pow((1.0 - t_2), c_n) / pow((1.0 - t_1), c_n);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    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
    real(8) :: tmp
    t_1 = 1.0d0 / (1.0d0 + exp(-t))
    t_2 = 1.0d0 / (1.0d0 + exp(-s))
    if (s <= (-3.7d+30)) then
        tmp = (t_2 ** c_p) / (t_1 ** c_p)
    else
        tmp = ((1.0d0 - t_2) ** c_n) / ((1.0d0 - t_1) ** c_n)
    end if
    code = tmp
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));
	double tmp;
	if (s <= -3.7e+30) {
		tmp = Math.pow(t_2, c_p) / Math.pow(t_1, c_p);
	} else {
		tmp = Math.pow((1.0 - t_2), c_n) / Math.pow((1.0 - t_1), c_n);
	}
	return tmp;
}

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))
	tmp = 0
	if s <= -3.7e+30:
		tmp = math.pow(t_2, c_p) / math.pow(t_1, c_p)
	else:
		tmp = math.pow((1.0 - t_2), c_n) / math.pow((1.0 - t_1), c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 / Float64(1.0 + exp(Float64(-t))))
	t_2 = Float64(1.0 / Float64(1.0 + exp(Float64(-s))))
	tmp = 0.0
	if (s <= -3.7e+30)
		tmp = Float64((t_2 ^ c_p) / (t_1 ^ c_p));
	else
		tmp = Float64((Float64(1.0 - t_2) ^ c_n) / (Float64(1.0 - t_1) ^ c_n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	t_1 = 1.0 / (1.0 + exp(-t));
	t_2 = 1.0 / (1.0 + exp(-s));
	tmp = 0.0;
	if (s <= -3.7e+30)
		tmp = (t_2 ^ c_p) / (t_1 ^ c_p);
	else
		tmp = ((1.0 - t_2) ^ c_n) / ((1.0 - t_1) ^ c_n);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1 / N[(1 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1 / N[(1 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s, -3700000000000000158015604916224], N[(N[Power[t$95$2, c$95$p], $MachinePrecision] / N[Power[t$95$1, c$95$p], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1 - t$95$2), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[N[(1 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if s < -3.7000000000000002e30

   1. Initial program 90.4%

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

   2. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

   4. Applied rewrites 92.6%

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

   if -3.7000000000000002e30 < s

   1. Initial program 90.4%

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

   2. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

   4. Applied rewrites 94.2%

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

Alternative 5: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{1}{1 + e^{-t}}\\ t_2 := {t\_1}^{c\_p}\\ t_3 := \frac{1}{1 + e^{-s}}\\ t_4 := {\left(1 - t\_3\right)}^{c\_n}\\ \mathbf{if}\;\frac{{t\_3}^{c\_p} \cdot t\_4}{t\_2 \cdot {\left(1 - t\_1\right)}^{c\_n}} \leq \frac{4503599852550477}{4503599627370496}:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\frac{1}{2}\right)}\\ \end{array} \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (let* ((t_1 (/ 1 (+ 1 (exp (- t)))))
       (t_2 (pow t_1 c_p))
       (t_3 (/ 1 (+ 1 (exp (- s)))))
       (t_4 (pow (- 1 t_3) c_n)))
  (if (<=
       (/ (* (pow t_3 c_p) t_4) (* t_2 (pow (- 1 t_1) c_n)))
       4503599852550477/4503599627370496)
    (/ (pow (/ 1 (+ 2 (* s (- (* 1/2 s) 1)))) c_p) t_2)
    (/ t_4 (+ 1 (* c_n (30-log1z0 1/2)))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n))) < 1.0000000499999999

   1. Initial program 90.4%

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

   2. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in s around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 93.4%

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

   7. Applied rewrites 93.4%

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

   if 1.0000000499999999 < (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n)))

   1. Initial program 90.4%

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

   2. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in c_n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-30-log1z0 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      7. lower-neg.f64 93.3%

         \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)} \]

   7. Applied rewrites 93.3%

      \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + \color{blue}{c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)}} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\frac{1}{2}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 93.7%

       \[\leadsto \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\frac{1}{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)\\ \mathbf{if}\;c\_n \leq 27000:\\ \;\;\;\;1 + c\_n \cdot \left(\mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-s}}\right)\right) - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot t\_1}\\ \end{array} \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (let* ((t_1 (30-log1z0 (/ 1 (+ 1 (exp (- t)))))))
  (if (<= c_n 27000)
    (+ 1 (* c_n (- (30-log1z0 (/ 1 (+ 1 (exp (- s))))) t_1)))
    (/ (pow 1/2 c_n) (+ 1 (* c_n t_1))))))

Derivation

1. Split input into 2 regimes

2. if c_n < 27000

   1. Initial program 90.4%

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

   2. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in c_n around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-30-log1z0 N/A

         \[\leadsto 1 + c\_n \cdot \left(\mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right) - \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto 1 + c\_n \cdot \left(\mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right) - \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + c\_n \cdot \left(\mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right) - \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right) \]

      7. lower-exp.f64 N/A

         \[\leadsto 1 + c\_n \cdot \left(\mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right) - \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto 1 + c\_n \cdot \left(\mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-s}}\right)\right) - \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right) \]

      9. lower-30-log1z0 N/A

         \[\leadsto 1 + c\_n \cdot \left(\mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-s}}\right)\right) - \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)\right) \]

   7. Applied rewrites 93.7%

      \[\leadsto 1 + \color{blue}{c\_n \cdot \left(\mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-s}}\right)\right) - \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)\right)} \]

   if 27000 < c_n

   1. Initial program 90.4%

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

   2. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in s around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-neg.f64 94.0%

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in c_n around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      3. lower-30-log1z0 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      7. lower-neg.f64 93.2%

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)} \]

   10. Applied rewrites 93.2%

       \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \color{blue}{\mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;c\_n \leq 27000:\\ \;\;\;\;1 + \frac{-1}{2} \cdot \left(c\_p \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)}\\ \end{array} \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (if (<= c_n 27000)
  (+ 1 (* -1/2 (* c_p t)))
  (/ (pow 1/2 c_n) (+ 1 (* c_n (30-log1z0 (/ 1 (+ 1 (exp (- t))))))))))

Derivation

1. Split input into 2 regimes

2. if c_n < 27000

   1. Initial program 90.4%

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

   2. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in s around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      7. lower-neg.f64 92.2%

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

   7. Applied rewrites 92.2%

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

   8. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{-1}{2} \cdot \left(c\_p \cdot t\right) \]

      3. lower-*.f64 94.4%

         \[\leadsto 1 + \frac{-1}{2} \cdot \left(c\_p \cdot t\right) \]

   10. Applied rewrites 94.4%

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

   if 27000 < c_n

   1. Initial program 90.4%

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

   2. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in s around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-neg.f64 94.0%

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

   7. Applied rewrites 94.0%

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

   8. Taylor expanded in c_n around 0

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

      1. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)} \]

      3. lower-30-log1z0 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)} \]

      7. lower-neg.f64 93.2%

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)} \]

   10. Applied rewrites 93.2%

       \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{1 + c\_n \cdot \color{blue}{\mathsf{30\_log1z0}\left(\left(\frac{1}{1 + e^{-t}}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.4% accurate, 4.0× speedup

Math

\[\frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (/ (pow 1/2 c_n) (pow (+ 1/2 (* -1/4 t)) c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	return pow(0.5, c_n) / pow((0.5 + (-0.25 * t)), c_n);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = (0.5d0 ** c_n) / ((0.5d0 + ((-0.25d0) * t)) ** c_n)
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow(0.5, c_n) / Math.pow((0.5 + (-0.25 * t)), c_n);
}

Python

def code(c_p, c_n, t, s):
	return math.pow(0.5, c_n) / math.pow((0.5 + (-0.25 * t)), c_n)

Julia

function code(c_p, c_n, t, s)
	return Float64((0.5 ^ c_n) / (Float64(0.5 + Float64(-0.25 * t)) ^ c_n))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = (0.5 ^ c_n) / ((0.5 + (-0.25 * t)) ^ c_n);
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[Power[1/2, c$95$n], $MachinePrecision] / N[Power[N[(1/2 + N[(-1/4 * t), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in c_p around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-pow.f64 N/A

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

4. Applied rewrites 94.2%

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

5. Taylor expanded in s around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   7. lower-exp.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   8. lower-neg.f64 94.0%

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

7. Applied rewrites 94.0%

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

8. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]

   2. lower-*.f64 93.7%

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]

10. Applied rewrites 93.7%

    \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]
11. Add Preprocessing

Alternative 9: 94.4% accurate, 4.1× speedup

Math

\[{\left(\frac{-1}{4} \cdot t - \frac{-1}{2}\right)}^{\left(-c\_n\right)} \cdot {\frac{1}{2}}^{c\_n} \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (* (pow (- (* -1/4 t) -1/2) (- c_n)) (pow 1/2 c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	return pow(((-0.25 * t) - -0.5), -c_n) * pow(0.5, c_n);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = ((((-0.25d0) * t) - (-0.5d0)) ** -c_n) * (0.5d0 ** c_n)
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow(((-0.25 * t) - -0.5), -c_n) * Math.pow(0.5, c_n);
}

Python

def code(c_p, c_n, t, s):
	return math.pow(((-0.25 * t) - -0.5), -c_n) * math.pow(0.5, c_n)

Julia

function code(c_p, c_n, t, s)
	return Float64((Float64(Float64(-0.25 * t) - -0.5) ^ Float64(-c_n)) * (0.5 ^ c_n))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = (((-0.25 * t) - -0.5) ^ -c_n) * (0.5 ^ c_n);
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(N[Power[N[(N[(-1/4 * t), $MachinePrecision] - -1/2), $MachinePrecision], (-c$95$n)], $MachinePrecision] * N[Power[1/2, c$95$n], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in c_p around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-pow.f64 N/A

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

4. Applied rewrites 94.2%

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

5. Taylor expanded in s around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   7. lower-exp.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   8. lower-neg.f64 94.0%

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

7. Applied rewrites 94.0%

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

8. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]

   2. lower-*.f64 93.7%

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]

10. Applied rewrites 93.7%

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

    1. lift-/.f64 N/A

       \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{\color{blue}{c\_n}}} \]

    2. mult-flip N/A

       \[\leadsto {\frac{1}{2}}^{c\_n} \cdot \frac{1}{\color{blue}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}}} \]

    3. *-commutative N/A

       \[\leadsto \frac{1}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \cdot {\frac{1}{2}}^{\color{blue}{c\_n}} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{1}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \cdot {\frac{1}{2}}^{\color{blue}{c\_n}} \]

    5. lift-pow.f64 N/A

       \[\leadsto \frac{1}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \cdot {\frac{1}{2}}^{c\_n} \]

    6. pow-flip N/A

       \[\leadsto {\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{\left(\mathsf{neg}\left(c\_n\right)\right)} \cdot {\frac{1}{2}}^{c\_n} \]

    7. lower-pow.f64 N/A

       \[\leadsto {\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{\left(\mathsf{neg}\left(c\_n\right)\right)} \cdot {\frac{1}{2}}^{c\_n} \]

    8. lift-+.f64 N/A

       \[\leadsto {\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{\left(\mathsf{neg}\left(c\_n\right)\right)} \cdot {\frac{1}{2}}^{c\_n} \]

    9. +-commutative N/A

       \[\leadsto {\left(\frac{-1}{4} \cdot t + \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(c\_n\right)\right)} \cdot {\frac{1}{2}}^{c\_n} \]

    10. add-flip N/A

        \[\leadsto {\left(\frac{-1}{4} \cdot t - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}^{\left(\mathsf{neg}\left(c\_n\right)\right)} \cdot {\frac{1}{2}}^{c\_n} \]

    11. lower--.f64 N/A

        \[\leadsto {\left(\frac{-1}{4} \cdot t - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}^{\left(\mathsf{neg}\left(c\_n\right)\right)} \cdot {\frac{1}{2}}^{c\_n} \]

    12. metadata-eval N/A

        \[\leadsto {\left(\frac{-1}{4} \cdot t - \frac{-1}{2}\right)}^{\left(\mathsf{neg}\left(c\_n\right)\right)} \cdot {\frac{1}{2}}^{c\_n} \]

    13. metadata-eval N/A

        \[\leadsto {\left(\frac{-1}{4} \cdot t - \frac{-1}{2}\right)}^{\mathsf{Rewrite=>}\left(lower-neg.f64, \left(-c\_n\right)\right)} \cdot {\frac{1}{2}}^{c\_n} \]

12. Applied rewrites 93.7%

    \[\leadsto {\left(\frac{-1}{4} \cdot t - \frac{-1}{2}\right)}^{\left(-c\_n\right)} \cdot {\frac{1}{2}}^{\color{blue}{c\_n}} \]
13. Add Preprocessing

Alternative 10: 93.7% accurate, 64.0× speedup

Math

\[1 + \frac{-1}{2} \cdot \left(c\_p \cdot t\right) \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (+ 1 (* -1/2 (* c_p t))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (-0.5 * (c_p * t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    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 + ((-0.5d0) * (c_p * t))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (-0.5 * (c_p * t));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + (-0.5 * (c_p * t))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 + Float64(-0.5 * Float64(c_p * t)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + (-0.5 * (c_p * t));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1 + N[(-1/2 * N[(c$95$p * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in c_n around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-pow.f64 N/A

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

4. Applied rewrites 92.6%

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

5. Taylor expanded in s around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   6. lower-exp.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

   7. lower-neg.f64 92.2%

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

7. Applied rewrites 92.2%

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

8. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

      \[\leadsto 1 + \frac{-1}{2} \cdot \left(c\_p \cdot t\right) \]

   3. lower-*.f64 94.4%

      \[\leadsto 1 + \frac{-1}{2} \cdot \left(c\_p \cdot t\right) \]

10. Applied rewrites 94.4%

    \[\leadsto 1 + \frac{-1}{2} \cdot \color{blue}{\left(c\_p \cdot t\right)} \]
11. Add Preprocessing

Alternative 11: 93.7% accurate, 64.0× speedup

Math

\[1 + \frac{1}{2} \cdot \left(c\_n \cdot t\right) \]

FPCore

(FPCore (c_p c_n t s)
  :precision binary64
  (+ 1 (* 1/2 (* c_n t))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (0.5 * (c_n * t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c_p, c_n, t, s)
use fmin_fmax_functions
    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 + (0.5d0 * (c_n * t))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (0.5 * (c_n * t));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + (0.5 * (c_n * t))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 + Float64(0.5 * Float64(c_n * t)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + (0.5 * (c_n * t));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1 + N[(1/2 * N[(c$95$n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in c_p around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-pow.f64 N/A

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

4. Applied rewrites 94.2%

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

5. Taylor expanded in s around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   6. lower-+.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   7. lower-exp.f64 N/A

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

   8. lower-neg.f64 94.0%

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

7. Applied rewrites 94.0%

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

8. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

      \[\leadsto 1 + \frac{1}{2} \cdot \left(c\_n \cdot t\right) \]

   3. lower-*.f64 94.4%

      \[\leadsto 1 + \frac{1}{2} \cdot \left(c\_n \cdot t\right) \]

10. Applied rewrites 94.4%

    \[\leadsto 1 + \frac{1}{2} \cdot \color{blue}{\left(c\_n \cdot t\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0 c_p) (< 0 c_n))
  (/ (* (pow (/ 1 (+ 1 (exp (- s)))) c_p) (pow (- 1 (/ 1 (+ 1 (exp (- s))))) c_n)) (* (pow (/ 1 (+ 1 (exp (- t)))) c_p) (pow (- 1 (/ 1 (+ 1 (exp (- t))))) c_n))))