Harley's example

Percentage Accurate: 91.5% → 97.7%

Time: 1.2min

Alternatives: 8

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 2.054741991264951e-75
  2. c_n = 3.539799752971644e+157
  3. t = -2.552010744431451e+26
  4. s = -4.6190864889538613e+136

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Initial Program: 91.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Alternative 1: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(e^{-s}\right)\\ t_2 := e^{\mathsf{fma}\left(\log \left(1 - e^{-t\_1}\right), c\_n, \left(-c\_p\right) \cdot t\_1\right) - \mathsf{fma}\left(-c\_p, \log 2, \log 0.5 \cdot c\_n\right)}\\ t_3 := \mathsf{fma}\left(c\_n, -0.5, 0.5 \cdot c\_p\right)\\ \mathbf{if}\;c\_p \leq 1.22 \cdot 10^{-54}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t\_2, \mathsf{fma}\left({t\_3}^{2}, 0.5, 0.125 \cdot \left(c\_p + c\_n\right)\right), \left(-t\_3\right) \cdot t\_2\right), t, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (log1p (exp (- s))))
        (t_2
         (exp
          (-
           (fma (log (- 1.0 (exp (- t_1)))) c_n (* (- c_p) t_1))
           (fma (- c_p) (log 2.0) (* (log 0.5) c_n)))))
        (t_3 (fma c_n -0.5 (* 0.5 c_p))))
   (if (<= c_p 1.22e-54)
     (/
      (pow (pow (fma (- (* 0.5 s) 1.0) s 2.0) -1.0) c_p)
      (fma (- c_p) (log1p (exp (- t))) 1.0))
     (fma
      (fma
       (* t t_2)
       (fma (pow t_3 2.0) 0.5 (* 0.125 (+ c_p c_n)))
       (* (- t_3) t_2))
      t
      t_2))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = log1p(exp(-s));
	double t_2 = exp((fma(log((1.0 - exp(-t_1))), c_n, (-c_p * t_1)) - fma(-c_p, log(2.0), (log(0.5) * c_n))));
	double t_3 = fma(c_n, -0.5, (0.5 * c_p));
	double tmp;
	if (c_p <= 1.22e-54) {
		tmp = pow(pow(fma(((0.5 * s) - 1.0), s, 2.0), -1.0), c_p) / fma(-c_p, log1p(exp(-t)), 1.0);
	} else {
		tmp = fma(fma((t * t_2), fma(pow(t_3, 2.0), 0.5, (0.125 * (c_p + c_n))), (-t_3 * t_2)), t, t_2);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = log1p(exp(Float64(-s)))
	t_2 = exp(Float64(fma(log(Float64(1.0 - exp(Float64(-t_1)))), c_n, Float64(Float64(-c_p) * t_1)) - fma(Float64(-c_p), log(2.0), Float64(log(0.5) * c_n))))
	t_3 = fma(c_n, -0.5, Float64(0.5 * c_p))
	tmp = 0.0
	if (c_p <= 1.22e-54)
		tmp = Float64(((fma(Float64(Float64(0.5 * s) - 1.0), s, 2.0) ^ -1.0) ^ c_p) / fma(Float64(-c_p), log1p(exp(Float64(-t))), 1.0));
	else
		tmp = fma(fma(Float64(t * t_2), fma((t_3 ^ 2.0), 0.5, Float64(0.125 * Float64(c_p + c_n))), Float64(Float64(-t_3) * t_2)), t, t_2);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(N[Log[N[(1.0 - N[Exp[(-t$95$1)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c$95$n + N[((-c$95$p) * t$95$1), $MachinePrecision]), $MachinePrecision] - N[((-c$95$p) * N[Log[2.0], $MachinePrecision] + N[(N[Log[0.5], $MachinePrecision] * c$95$n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(c$95$n * -0.5 + N[(0.5 * c$95$p), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$p, 1.22e-54], N[(N[Power[N[Power[N[(N[(N[(0.5 * s), $MachinePrecision] - 1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], -1.0], $MachinePrecision], c$95$p], $MachinePrecision] / N[((-c$95$p) * N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * t$95$2), $MachinePrecision] * N[(N[Power[t$95$3, 2.0], $MachinePrecision] * 0.5 + N[(0.125 * N[(c$95$p + c$95$n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-t$95$3) * t$95$2), $MachinePrecision]), $MachinePrecision] * t + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1.22e-54

   1. Initial program 91.3%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 95.8%

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

   9. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.9%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]

   if 1.22e-54 < c_p

   1. Initial program 75.4%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in t around 0

      \[\leadsto \color{blue}{e^{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right) + c\_n \cdot \log \left(1 - e^{\mathsf{neg}\left(\log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)}\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_n \cdot \log \left(1 - e^{\mathsf{neg}\left(\log 2\right)}\right)\right)} + t \cdot \left(-1 \cdot \left(e^{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right) + c\_n \cdot \log \left(1 - e^{\mathsf{neg}\left(\log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)}\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_n \cdot \log \left(1 - e^{\mathsf{neg}\left(\log 2\right)}\right)\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c\_n \cdot e^{\mathsf{neg}\left(\log 2\right)}}{1 - e^{\mathsf{neg}\left(\log 2\right)}} + \frac{1}{2} \cdot c\_p\right)\right) + t \cdot \left(e^{\left(-1 \cdot \left(c\_p \cdot \log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right) + c\_n \cdot \log \left(1 - e^{\mathsf{neg}\left(\log \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)}\right)\right) - \left(-1 \cdot \left(c\_p \cdot \log 2\right) + c\_n \cdot \log \left(1 - e^{\mathsf{neg}\left(\log 2\right)}\right)\right)} \cdot \left(-1 \cdot \left(\frac{-1}{8} \cdot c\_p + \frac{-1}{8} \cdot \frac{c\_n \cdot {\left(e^{\mathsf{neg}\left(\log 2\right)}\right)}^{2}}{{\left(1 - e^{\mathsf{neg}\left(\log 2\right)}\right)}^{2}}\right) + \frac{1}{2} \cdot {\left(\frac{-1}{2} \cdot \frac{c\_n \cdot e^{\mathsf{neg}\left(\log 2\right)}}{1 - e^{\mathsf{neg}\left(\log 2\right)}} + \frac{1}{2} \cdot c\_p\right)}^{2}\right)\right)\right)} \]

   7. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t \cdot e^{\mathsf{fma}\left(\log \left(1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}\right), c\_n, \left(-c\_p\right) \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(-c\_p, \log 2, \log 0.5 \cdot c\_n\right)}, \mathsf{fma}\left({\left(\mathsf{fma}\left(c\_n \cdot 1, -0.5, 0.5 \cdot c\_p\right)\right)}^{2}, 0.5, 0.125 \cdot \left(c\_p + c\_n \cdot 1\right)\right), \left(-\mathsf{fma}\left(c\_n \cdot 1, -0.5, 0.5 \cdot c\_p\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}\right), c\_n, \left(-c\_p\right) \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(-c\_p, \log 2, \log 0.5 \cdot c\_n\right)}\right), t, e^{\mathsf{fma}\left(\log \left(1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}\right), c\_n, \left(-c\_p\right) \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(-c\_p, \log 2, \log 0.5 \cdot c\_n\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 1.22 \cdot 10^{-54}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t \cdot e^{\mathsf{fma}\left(\log \left(1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}\right), c\_n, \left(-c\_p\right) \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(-c\_p, \log 2, \log 0.5 \cdot c\_n\right)}, \mathsf{fma}\left({\left(\mathsf{fma}\left(c\_n, -0.5, 0.5 \cdot c\_p\right)\right)}^{2}, 0.5, 0.125 \cdot \left(c\_p + c\_n\right)\right), \left(-\mathsf{fma}\left(c\_n, -0.5, 0.5 \cdot c\_p\right)\right) \cdot e^{\mathsf{fma}\left(\log \left(1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}\right), c\_n, \left(-c\_p\right) \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(-c\_p, \log 2, \log 0.5 \cdot c\_n\right)}\right), t, e^{\mathsf{fma}\left(\log \left(1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}\right), c\_n, \left(-c\_p\right) \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - \mathsf{fma}\left(-c\_p, \log 2, \log 0.5 \cdot c\_n\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(e^{-t}\right)\\ t_2 := -t\_1\\ t_3 := -\mathsf{log1p}\left(e^{-s}\right)\\ \mathbf{if}\;c\_p \leq 1.24 \cdot 10^{-54}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, t\_1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(t\_3, c\_p, \mathsf{log1p}\left(-e^{t\_3}\right) \cdot c\_n\right) - \mathsf{fma}\left(t\_2, c\_p, \mathsf{log1p}\left(-e^{t\_2}\right) \cdot c\_n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (log1p (exp (- t)))) (t_2 (- t_1)) (t_3 (- (log1p (exp (- s))))))
   (if (<= c_p 1.24e-54)
     (/
      (pow (pow (fma (- (* 0.5 s) 1.0) s 2.0) -1.0) c_p)
      (fma (- c_p) t_1 1.0))
     (exp
      (-
       (fma t_3 c_p (* (log1p (- (exp t_3))) c_n))
       (fma t_2 c_p (* (log1p (- (exp t_2))) c_n)))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = log1p(exp(-t));
	double t_2 = -t_1;
	double t_3 = -log1p(exp(-s));
	double tmp;
	if (c_p <= 1.24e-54) {
		tmp = pow(pow(fma(((0.5 * s) - 1.0), s, 2.0), -1.0), c_p) / fma(-c_p, t_1, 1.0);
	} else {
		tmp = exp((fma(t_3, c_p, (log1p(-exp(t_3)) * c_n)) - fma(t_2, c_p, (log1p(-exp(t_2)) * c_n))));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = log1p(exp(Float64(-t)))
	t_2 = Float64(-t_1)
	t_3 = Float64(-log1p(exp(Float64(-s))))
	tmp = 0.0
	if (c_p <= 1.24e-54)
		tmp = Float64(((fma(Float64(Float64(0.5 * s) - 1.0), s, 2.0) ^ -1.0) ^ c_p) / fma(Float64(-c_p), t_1, 1.0));
	else
		tmp = exp(Float64(fma(t_3, c_p, Float64(log1p(Float64(-exp(t_3))) * c_n)) - fma(t_2, c_p, Float64(log1p(Float64(-exp(t_2))) * c_n))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, Block[{t$95$3 = (-N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision])}, If[LessEqual[c$95$p, 1.24e-54], N[(N[Power[N[Power[N[(N[(N[(0.5 * s), $MachinePrecision] - 1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], -1.0], $MachinePrecision], c$95$p], $MachinePrecision] / N[((-c$95$p) * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(t$95$3 * c$95$p + N[(N[Log[1 + (-N[Exp[t$95$3], $MachinePrecision])], $MachinePrecision] * c$95$n), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * c$95$p + N[(N[Log[1 + (-N[Exp[t$95$2], $MachinePrecision])], $MachinePrecision] * c$95$n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1.23999999999999999e-54

   1. Initial program 91.3%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 95.8%

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

   9. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.9%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]

   if 1.23999999999999999e-54 < c_p

   1. Initial program 75.4%

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

   4. Applied rewrites 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 1.24 \cdot 10^{-54}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-s}\right), c\_p, \mathsf{log1p}\left(-e^{-\mathsf{log1p}\left(e^{-s}\right)}\right) \cdot c\_n\right) - \mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, \mathsf{log1p}\left(-e^{-\mathsf{log1p}\left(e^{-t}\right)}\right) \cdot c\_n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-s}\right) - \log 2, \left(t \cdot c\_p\right) \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 2e-40)
   (/
    (pow (pow (fma (- (* 0.5 s) 1.0) s 2.0) -1.0) c_p)
    (fma (- c_p) (log1p (exp (- t))) 1.0))
   (exp (fma (- c_p) (- (log1p (exp (- s))) (log 2.0)) (* (* t c_p) -0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 2e-40) {
		tmp = pow(pow(fma(((0.5 * s) - 1.0), s, 2.0), -1.0), c_p) / fma(-c_p, log1p(exp(-t)), 1.0);
	} else {
		tmp = exp(fma(-c_p, (log1p(exp(-s)) - log(2.0)), ((t * c_p) * -0.5)));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 2e-40)
		tmp = Float64(((fma(Float64(Float64(0.5 * s) - 1.0), s, 2.0) ^ -1.0) ^ c_p) / fma(Float64(-c_p), log1p(exp(Float64(-t))), 1.0));
	else
		tmp = exp(fma(Float64(-c_p), Float64(log1p(exp(Float64(-s))) - log(2.0)), Float64(Float64(t * c_p) * -0.5)));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 2e-40], N[(N[Power[N[Power[N[(N[(N[(0.5 * s), $MachinePrecision] - 1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], -1.0], $MachinePrecision], c$95$p], $MachinePrecision] / N[((-c$95$p) * N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[N[((-c$95$p) * N[(N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(t * c$95$p), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1.9999999999999999e-40

   1. Initial program 91.4%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 95.9%

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

   9. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.7%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]

   if 1.9999999999999999e-40 < c_p

   1. Initial program 70.6%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 97.8%

      \[\leadsto e^{\mathsf{log1p}\left(e^{-s}\right) \cdot \left(-c\_p\right) - \mathsf{log1p}\left(e^{-t}\right) \cdot \left(-c\_p\right)} \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 97.9%

       \[\leadsto e^{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-s}\right) - \log 2, \left(t \cdot c\_p\right) \cdot -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-s}\right) - \log 2, \left(t \cdot c\_p\right) \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(e^{-s}\right) \cdot \left(-c\_p\right) + \mathsf{log1p}\left(1 - t\right) \cdot c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 2e-40)
   (/
    (pow (pow (fma (- (* 0.5 s) 1.0) s 2.0) -1.0) c_p)
    (fma (- c_p) (log1p (exp (- t))) 1.0))
   (exp (+ (* (log1p (exp (- s))) (- c_p)) (* (log1p (- 1.0 t)) c_p)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 2e-40) {
		tmp = pow(pow(fma(((0.5 * s) - 1.0), s, 2.0), -1.0), c_p) / fma(-c_p, log1p(exp(-t)), 1.0);
	} else {
		tmp = exp(((log1p(exp(-s)) * -c_p) + (log1p((1.0 - t)) * c_p)));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 2e-40)
		tmp = Float64(((fma(Float64(Float64(0.5 * s) - 1.0), s, 2.0) ^ -1.0) ^ c_p) / fma(Float64(-c_p), log1p(exp(Float64(-t))), 1.0));
	else
		tmp = exp(Float64(Float64(log1p(exp(Float64(-s))) * Float64(-c_p)) + Float64(log1p(Float64(1.0 - t)) * c_p)));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 2e-40], N[(N[Power[N[Power[N[(N[(N[(0.5 * s), $MachinePrecision] - 1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], -1.0], $MachinePrecision], c$95$p], $MachinePrecision] / N[((-c$95$p) * N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision] * (-c$95$p)), $MachinePrecision] + N[(N[Log[1 + N[(1.0 - t), $MachinePrecision]], $MachinePrecision] * c$95$p), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1.9999999999999999e-40

   1. Initial program 91.4%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 95.9%

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

   9. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.7%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]

   if 1.9999999999999999e-40 < c_p

   1. Initial program 70.6%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 97.8%

      \[\leadsto e^{\mathsf{log1p}\left(e^{-s}\right) \cdot \left(-c\_p\right) - \mathsf{log1p}\left(e^{-t}\right) \cdot \left(-c\_p\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto e^{\mathsf{log1p}\left(e^{-s}\right) \cdot \left(-c\_p\right) - \mathsf{log1p}\left(1 + -1 \cdot t\right) \cdot \left(-c\_p\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(e^{-s}\right) \cdot \left(-c\_p\right) + \mathsf{log1p}\left(1 - t\right) \cdot c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-t}\\ \mathbf{if}\;c\_p \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{e^{-s} + 1}{t\_1 + 1}\right)}^{\left(-c\_p\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- t))))
   (if (<= c_p 2e-40)
     (/
      (pow (pow (fma (- (* 0.5 s) 1.0) s 2.0) -1.0) c_p)
      (fma (- c_p) (log1p t_1) 1.0))
     (pow (/ (+ (exp (- s)) 1.0) (+ t_1 1.0)) (- c_p)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-t);
	double tmp;
	if (c_p <= 2e-40) {
		tmp = pow(pow(fma(((0.5 * s) - 1.0), s, 2.0), -1.0), c_p) / fma(-c_p, log1p(t_1), 1.0);
	} else {
		tmp = pow(((exp(-s) + 1.0) / (t_1 + 1.0)), -c_p);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-t))
	tmp = 0.0
	if (c_p <= 2e-40)
		tmp = Float64(((fma(Float64(Float64(0.5 * s) - 1.0), s, 2.0) ^ -1.0) ^ c_p) / fma(Float64(-c_p), log1p(t_1), 1.0));
	else
		tmp = Float64(Float64(exp(Float64(-s)) + 1.0) / Float64(t_1 + 1.0)) ^ Float64(-c_p);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-t)], $MachinePrecision]}, If[LessEqual[c$95$p, 2e-40], N[(N[Power[N[Power[N[(N[(N[(0.5 * s), $MachinePrecision] - 1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], -1.0], $MachinePrecision], c$95$p], $MachinePrecision] / N[((-c$95$p) * N[Log[1 + t$95$1], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], (-c$95$p)], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1.9999999999999999e-40

   1. Initial program 91.4%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 95.9%

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

   9. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.7%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]

   if 1.9999999999999999e-40 < c_p

   1. Initial program 70.6%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 97.8%

      \[\leadsto e^{\mathsf{log1p}\left(e^{-s}\right) \cdot \left(-c\_p\right) - \mathsf{log1p}\left(e^{-t}\right) \cdot \left(-c\_p\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{e^{-s} + 1}{e^{-t} + 1}\right)}^{\left(-c\_p\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-t}\\ \mathbf{if}\;c\_p \leq 1.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{{\left({2}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(t\_1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{e^{-s} + 1}{t\_1 + 1}\right)}^{\left(-c\_p\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- t))))
   (if (<= c_p 1.5e-54)
     (/ (pow (pow 2.0 -1.0) c_p) (fma (- c_p) (log1p t_1) 1.0))
     (pow (/ (+ (exp (- s)) 1.0) (+ t_1 1.0)) (- c_p)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-t);
	double tmp;
	if (c_p <= 1.5e-54) {
		tmp = pow(pow(2.0, -1.0), c_p) / fma(-c_p, log1p(t_1), 1.0);
	} else {
		tmp = pow(((exp(-s) + 1.0) / (t_1 + 1.0)), -c_p);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-t))
	tmp = 0.0
	if (c_p <= 1.5e-54)
		tmp = Float64(((2.0 ^ -1.0) ^ c_p) / fma(Float64(-c_p), log1p(t_1), 1.0));
	else
		tmp = Float64(Float64(exp(Float64(-s)) + 1.0) / Float64(t_1 + 1.0)) ^ Float64(-c_p);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-t)], $MachinePrecision]}, If[LessEqual[c$95$p, 1.5e-54], N[(N[Power[N[Power[2.0, -1.0], $MachinePrecision], c$95$p], $MachinePrecision] / N[((-c$95$p) * N[Log[1 + t$95$1], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], (-c$95$p)], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1.50000000000000005e-54

   1. Initial program 91.3%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 94.3

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 95.8%

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

   9. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.4%

       \[\leadsto \frac{{\left(\frac{1}{2}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)} \]

   if 1.50000000000000005e-54 < c_p

   1. Initial program 75.4%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 97.5%

      \[\leadsto e^{\mathsf{log1p}\left(e^{-s}\right) \cdot \left(-c\_p\right) - \mathsf{log1p}\left(e^{-t}\right) \cdot \left(-c\_p\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 1.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{{\left({2}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-c\_p, \mathsf{log1p}\left(e^{-t}\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{e^{-s} + 1}{e^{-t} + 1}\right)}^{\left(-c\_p\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{e^{-s} + 1}{e^{-t} + 1}\right)}^{\left(-c\_p\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 2e-40)
   1.0
   (pow (/ (+ (exp (- s)) 1.0) (+ (exp (- t)) 1.0)) (- c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 2e-40) {
		tmp = 1.0;
	} else {
		tmp = pow(((exp(-s) + 1.0) / (exp(-t) + 1.0)), -c_p);
	}
	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) :: tmp
    if (c_p <= 2d-40) then
        tmp = 1.0d0
    else
        tmp = ((exp(-s) + 1.0d0) / (exp(-t) + 1.0d0)) ** -c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 2e-40) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(((Math.exp(-s) + 1.0) / (Math.exp(-t) + 1.0)), -c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 2e-40:
		tmp = 1.0
	else:
		tmp = math.pow(((math.exp(-s) + 1.0) / (math.exp(-t) + 1.0)), -c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 2e-40)
		tmp = 1.0;
	else
		tmp = Float64(Float64(exp(Float64(-s)) + 1.0) / Float64(exp(Float64(-t)) + 1.0)) ^ Float64(-c_p);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_p <= 2e-40)
		tmp = 1.0;
	else
		tmp = ((exp(-s) + 1.0) / (exp(-t) + 1.0)) ^ -c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 2e-40], 1.0, N[Power[N[(N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Exp[(-t)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], (-c$95$p)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1.9999999999999999e-40

   1. Initial program 91.4%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 97.3%

      \[\leadsto 1 \]

   if 1.9999999999999999e-40 < c_p

   1. Initial program 70.6%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 97.8%

      \[\leadsto e^{\mathsf{log1p}\left(e^{-s}\right) \cdot \left(-c\_p\right) - \mathsf{log1p}\left(e^{-t}\right) \cdot \left(-c\_p\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.6%

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

Alternative 8: 94.4% accurate, 896.0× speedup

Math

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

FPCore

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

C

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

Fortran

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

3. Taylor expanded in c_n around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 90.5

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

5. Applied rewrites 90.5%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 94.7%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 96.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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 + exp(-t)) / (1.0d0 + exp(-s))) ** c_p) * (((1.0d0 + exp(t)) / (1.0d0 + exp(s))) ** c_n)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024351 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

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

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