Harley's example

Percentage Accurate: 90.8% → 97.9%

Time: 2.3min

Alternatives: 6

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 1.7681547809745393e+161
  2. c_n = 1.75494575491682e-5
  3. t = 3.014503680142846e+119
  4. s = 3.460809179524717e+197

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 200:\\ \;\;\;\;{\left(e^{c\_n}\right)}^{\log \left(\frac{1 + \frac{-1}{1 + e^{-s}}}{1 + \frac{-1}{1 + e^{-t}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{-6}{{t}^{3}}\right)}^{c\_p}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 200.0)
   (pow
    (exp c_n)
    (log
     (/
      (+ 1.0 (/ -1.0 (+ 1.0 (exp (- s)))))
      (+ 1.0 (/ -1.0 (+ 1.0 (exp (- t))))))))
   (/ 1.0 (pow (/ -6.0 (pow t 3.0)) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 200.0) {
		tmp = pow(exp(c_n), log(((1.0 + (-1.0 / (1.0 + exp(-s)))) / (1.0 + (-1.0 / (1.0 + exp(-t)))))));
	} else {
		tmp = 1.0 / pow((-6.0 / pow(t, 3.0)), c_p);
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (-s <= 200.0d0) then
        tmp = exp(c_n) ** log(((1.0d0 + ((-1.0d0) / (1.0d0 + exp(-s)))) / (1.0d0 + ((-1.0d0) / (1.0d0 + exp(-t))))))
    else
        tmp = 1.0d0 / (((-6.0d0) / (t ** 3.0d0)) ** 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 (-s <= 200.0) {
		tmp = Math.pow(Math.exp(c_n), Math.log(((1.0 + (-1.0 / (1.0 + Math.exp(-s)))) / (1.0 + (-1.0 / (1.0 + Math.exp(-t)))))));
	} else {
		tmp = 1.0 / Math.pow((-6.0 / Math.pow(t, 3.0)), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 200.0:
		tmp = math.pow(math.exp(c_n), math.log(((1.0 + (-1.0 / (1.0 + math.exp(-s)))) / (1.0 + (-1.0 / (1.0 + math.exp(-t)))))))
	else:
		tmp = 1.0 / math.pow((-6.0 / math.pow(t, 3.0)), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 200.0)
		tmp = exp(c_n) ^ log(Float64(Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(Float64(-s))))) / Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(Float64(-t)))))));
	else
		tmp = Float64(1.0 / (Float64(-6.0 / (t ^ 3.0)) ^ c_p));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 200.0)
		tmp = exp(c_n) ^ log(((1.0 + (-1.0 / (1.0 + exp(-s)))) / (1.0 + (-1.0 / (1.0 + exp(-t))))));
	else
		tmp = 1.0 / ((-6.0 / (t ^ 3.0)) ^ c_p);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 200.0], N[Power[N[Exp[c$95$n], $MachinePrecision], N[Log[N[(N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Power[N[(-6.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 200

   1. Initial program 92.9%

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

      1. associate-/l/ 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in c_p around 0 96.4%

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

      1. expm1-log1p-u 96.4%

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

   7. Applied egg-rr 96.4%

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

   8. Taylor expanded in s around inf 96.4%

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

   10. Simplified 99.6%

       \[\leadsto \color{blue}{{\left(e^{c\_n}\right)}^{\left(\mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)\right)}} \]
   11. Step-by-step derivation

       1. log1p-undefine 99.6%

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

       2. log1p-undefine 99.6%

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

       3. diff-log 99.6%

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

   12. Applied egg-rr 99.6%

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

   if 200 < (neg.f64 s)

   1. Initial program 16.7%

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

      1. associate-/l/ 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in c_n around 0 16.7%

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

   6. Taylor expanded in t around 0 16.7%

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

   7. Taylor expanded in c_p around 0 1.8%

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

   8. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{1}{{\color{blue}{\left(\frac{-6}{{t}^{3}}\right)}}^{c\_p}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 95.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\frac{{\left(1 + e^{s}\right)}^{\left(-c\_p\right)}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{0.5}^{c\_p} + 0.5 \cdot \left(t \cdot c\_p\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) -5e+69)
   (/ (pow (+ 1.0 (exp s)) (- c_p)) (pow 0.5 c_p))
   (/
    (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p)
    (+ (pow 0.5 c_p) (* 0.5 (* t c_p))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= -5e+69) {
		tmp = pow((1.0 + exp(s)), -c_p) / pow(0.5, c_p);
	} else {
		tmp = pow((1.0 / (1.0 + exp(-s))), c_p) / (pow(0.5, c_p) + (0.5 * (t * c_p)));
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (-s <= (-5d+69)) then
        tmp = ((1.0d0 + exp(s)) ** -c_p) / (0.5d0 ** c_p)
    else
        tmp = ((1.0d0 / (1.0d0 + exp(-s))) ** c_p) / ((0.5d0 ** c_p) + (0.5d0 * (t * c_p)))
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= -5e+69) {
		tmp = Math.pow((1.0 + Math.exp(s)), -c_p) / Math.pow(0.5, c_p);
	} else {
		tmp = Math.pow((1.0 / (1.0 + Math.exp(-s))), c_p) / (Math.pow(0.5, c_p) + (0.5 * (t * c_p)));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= -5e+69:
		tmp = math.pow((1.0 + math.exp(s)), -c_p) / math.pow(0.5, c_p)
	else:
		tmp = math.pow((1.0 / (1.0 + math.exp(-s))), c_p) / (math.pow(0.5, c_p) + (0.5 * (t * c_p)))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= -5e+69)
		tmp = Float64((Float64(1.0 + exp(s)) ^ Float64(-c_p)) / (0.5 ^ c_p));
	else
		tmp = Float64((Float64(1.0 / Float64(1.0 + exp(Float64(-s)))) ^ c_p) / Float64((0.5 ^ c_p) + Float64(0.5 * Float64(t * c_p))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= -5e+69)
		tmp = ((1.0 + exp(s)) ^ -c_p) / (0.5 ^ c_p);
	else
		tmp = ((1.0 / (1.0 + exp(-s))) ^ c_p) / ((0.5 ^ c_p) + (0.5 * (t * c_p)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), -5e+69], N[(N[Power[N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] / N[(N[Power[0.5, c$95$p], $MachinePrecision] + N[(0.5 * N[(t * c$95$p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < -5.00000000000000036e69

   1. Initial program 66.7%

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

      1. associate-/l/ 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in c_n around 0 2.9%

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

   6. Taylor expanded in t around 0 3.1%

      \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{2 + t \cdot \left(t \cdot \left(0.5 + -0.16666666666666666 \cdot t\right) - 1\right)}}\right)}^{c\_p}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 3.1%

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

      2. inv-pow 3.1%

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

      3. pow-pow 3.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 100.0%

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

      6. sqr-neg 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{1 \cdot {\left(1 + e^{s}\right)}^{\left(-1 \cdot c\_p\right)}}}{{\left(\frac{1}{2 + t \cdot \left(t \cdot \left(0.5 + -0.16666666666666666 \cdot t\right) - 1\right)}\right)}^{c\_p}} \]
   9. Step-by-step derivation

      1. *-lft-identity 100.0%

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

      2. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in t around 0 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

       4. exp-to-pow 100.0%

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

   13. Simplified 100.0%

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

   if -5.00000000000000036e69 < (neg.f64 s)

   1. Initial program 91.7%

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

      1. associate-/l/ 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in t around 0 94.2%

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

   6. Taylor expanded in c_n around 0 95.7%

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

   7. Taylor expanded in c_p around 0 97.8%

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

      1. *-commutative 97.8%

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

   9. Simplified 97.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\frac{{\left(1 + e^{s}\right)}^{\left(-c\_p\right)}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{0.5}^{c\_p} + 0.5 \cdot \left(t \cdot c\_p\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq -5 \cdot 10^{+69}:\\ \;\;\;\;\frac{{\left(1 + e^{s}\right)}^{\left(-c\_p\right)}}{{0.5}^{c\_p}}\\ \mathbf{elif}\;-s \leq 200:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{-6}{{t}^{3}}\right)}^{c\_p}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) -5e+69)
   (/ (pow (+ 1.0 (exp s)) (- c_p)) (pow 0.5 c_p))
   (if (<= (- s) 200.0) 1.0 (/ 1.0 (pow (/ -6.0 (pow t 3.0)) c_p)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= -5e+69) {
		tmp = pow((1.0 + exp(s)), -c_p) / pow(0.5, c_p);
	} else if (-s <= 200.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / pow((-6.0 / pow(t, 3.0)), c_p);
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (-s <= (-5d+69)) then
        tmp = ((1.0d0 + exp(s)) ** -c_p) / (0.5d0 ** c_p)
    else if (-s <= 200.0d0) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (((-6.0d0) / (t ** 3.0d0)) ** 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 (-s <= -5e+69) {
		tmp = Math.pow((1.0 + Math.exp(s)), -c_p) / Math.pow(0.5, c_p);
	} else if (-s <= 200.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / Math.pow((-6.0 / Math.pow(t, 3.0)), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= -5e+69:
		tmp = math.pow((1.0 + math.exp(s)), -c_p) / math.pow(0.5, c_p)
	elif -s <= 200.0:
		tmp = 1.0
	else:
		tmp = 1.0 / math.pow((-6.0 / math.pow(t, 3.0)), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= -5e+69)
		tmp = Float64((Float64(1.0 + exp(s)) ^ Float64(-c_p)) / (0.5 ^ c_p));
	elseif (Float64(-s) <= 200.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / (Float64(-6.0 / (t ^ 3.0)) ^ c_p));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= -5e+69)
		tmp = ((1.0 + exp(s)) ^ -c_p) / (0.5 ^ c_p);
	elseif (-s <= 200.0)
		tmp = 1.0;
	else
		tmp = 1.0 / ((-6.0 / (t ^ 3.0)) ^ c_p);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), -5e+69], N[(N[Power[N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision], If[LessEqual[(-s), 200.0], 1.0, N[(1.0 / N[Power[N[(-6.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (neg.f64 s) < -5.00000000000000036e69

   1. Initial program 66.7%

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

      1. associate-/l/ 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in c_n around 0 2.9%

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

   6. Taylor expanded in t around 0 3.1%

      \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{2 + t \cdot \left(t \cdot \left(0.5 + -0.16666666666666666 \cdot t\right) - 1\right)}}\right)}^{c\_p}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 3.1%

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

      2. inv-pow 3.1%

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

      3. pow-pow 3.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 100.0%

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

      6. sqr-neg 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{1 \cdot {\left(1 + e^{s}\right)}^{\left(-1 \cdot c\_p\right)}}}{{\left(\frac{1}{2 + t \cdot \left(t \cdot \left(0.5 + -0.16666666666666666 \cdot t\right) - 1\right)}\right)}^{c\_p}} \]
   9. Step-by-step derivation

      1. *-lft-identity 100.0%

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

      2. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in t around 0 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

       4. exp-to-pow 100.0%

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

   13. Simplified 100.0%

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

   if -5.00000000000000036e69 < (neg.f64 s) < 200

   1. Initial program 93.5%

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

      1. associate-/l/ 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in c_p around 0 96.8%

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

   6. Taylor expanded in c_n around 0 98.3%

      \[\leadsto \color{blue}{1} \]

   if 200 < (neg.f64 s)

   1. Initial program 16.7%

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

      1. associate-/l/ 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in c_n around 0 16.7%

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

   6. Taylor expanded in t around 0 16.7%

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

   7. Taylor expanded in c_p around 0 1.8%

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

   8. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{1}{{\color{blue}{\left(\frac{-6}{{t}^{3}}\right)}}^{c\_p}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 6.2:\\ \;\;\;\;\frac{{\left(2 + s \cdot \left(1 + s \cdot \left(0.5 + s \cdot 0.16666666666666666\right)\right)\right)}^{\left(-c\_p\right)}}{{\left(\frac{1}{2 + t \cdot \left(-1 + t \cdot \left(0.5 + t \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{-6}{{t}^{3}}\right)}^{c\_p}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 6.2)
   (/
    (pow (+ 2.0 (* s (+ 1.0 (* s (+ 0.5 (* s 0.16666666666666666)))))) (- c_p))
    (pow
     (/ 1.0 (+ 2.0 (* t (+ -1.0 (* t (+ 0.5 (* t -0.16666666666666666)))))))
     c_p))
   (/ 1.0 (pow (/ -6.0 (pow t 3.0)) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 6.2) {
		tmp = pow((2.0 + (s * (1.0 + (s * (0.5 + (s * 0.16666666666666666)))))), -c_p) / pow((1.0 / (2.0 + (t * (-1.0 + (t * (0.5 + (t * -0.16666666666666666))))))), c_p);
	} else {
		tmp = 1.0 / pow((-6.0 / pow(t, 3.0)), c_p);
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (c_p <= 6.2d0) then
        tmp = ((2.0d0 + (s * (1.0d0 + (s * (0.5d0 + (s * 0.16666666666666666d0)))))) ** -c_p) / ((1.0d0 / (2.0d0 + (t * ((-1.0d0) + (t * (0.5d0 + (t * (-0.16666666666666666d0)))))))) ** c_p)
    else
        tmp = 1.0d0 / (((-6.0d0) / (t ** 3.0d0)) ** 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 <= 6.2) {
		tmp = Math.pow((2.0 + (s * (1.0 + (s * (0.5 + (s * 0.16666666666666666)))))), -c_p) / Math.pow((1.0 / (2.0 + (t * (-1.0 + (t * (0.5 + (t * -0.16666666666666666))))))), c_p);
	} else {
		tmp = 1.0 / Math.pow((-6.0 / Math.pow(t, 3.0)), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 6.2:
		tmp = math.pow((2.0 + (s * (1.0 + (s * (0.5 + (s * 0.16666666666666666)))))), -c_p) / math.pow((1.0 / (2.0 + (t * (-1.0 + (t * (0.5 + (t * -0.16666666666666666))))))), c_p)
	else:
		tmp = 1.0 / math.pow((-6.0 / math.pow(t, 3.0)), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 6.2)
		tmp = Float64((Float64(2.0 + Float64(s * Float64(1.0 + Float64(s * Float64(0.5 + Float64(s * 0.16666666666666666)))))) ^ Float64(-c_p)) / (Float64(1.0 / Float64(2.0 + Float64(t * Float64(-1.0 + Float64(t * Float64(0.5 + Float64(t * -0.16666666666666666))))))) ^ c_p));
	else
		tmp = Float64(1.0 / (Float64(-6.0 / (t ^ 3.0)) ^ c_p));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_p <= 6.2)
		tmp = ((2.0 + (s * (1.0 + (s * (0.5 + (s * 0.16666666666666666)))))) ^ -c_p) / ((1.0 / (2.0 + (t * (-1.0 + (t * (0.5 + (t * -0.16666666666666666))))))) ^ c_p);
	else
		tmp = 1.0 / ((-6.0 / (t ^ 3.0)) ^ c_p);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 6.2], N[(N[Power[N[(2.0 + N[(s * N[(1.0 + N[(s * N[(0.5 + N[(s * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[N[(1.0 / N[(2.0 + N[(t * N[(-1.0 + N[(t * N[(0.5 + N[(t * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(-6.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 6.20000000000000018

   1. Initial program 93.6%

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

      1. associate-/l/ 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in c_n around 0 94.4%

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

   6. Taylor expanded in t around 0 96.2%

      \[\leadsto \frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}}{{\left(\frac{1}{\color{blue}{2 + t \cdot \left(t \cdot \left(0.5 + -0.16666666666666666 \cdot t\right) - 1\right)}}\right)}^{c\_p}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 96.2%

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

      2. inv-pow 96.2%

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

      3. pow-pow 96.2%

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

      4. add-sqr-sqrt 46.7%

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

      5. sqrt-unprod 97.2%

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

      6. sqr-neg 97.2%

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

      7. sqrt-unprod 50.5%

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

      8. add-sqr-sqrt 97.2%

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

   8. Applied egg-rr 97.2%

      \[\leadsto \frac{\color{blue}{1 \cdot {\left(1 + e^{s}\right)}^{\left(-1 \cdot c\_p\right)}}}{{\left(\frac{1}{2 + t \cdot \left(t \cdot \left(0.5 + -0.16666666666666666 \cdot t\right) - 1\right)}\right)}^{c\_p}} \]
   9. Step-by-step derivation

      1. *-lft-identity 97.2%

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

      2. neg-mul-1 97.2%

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

   10. Simplified 97.2%

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

   11. Taylor expanded in s around 0 97.4%

       \[\leadsto \frac{{\color{blue}{\left(2 + s \cdot \left(1 + s \cdot \left(0.5 + 0.16666666666666666 \cdot s\right)\right)\right)}}^{\left(-c\_p\right)}}{{\left(\frac{1}{2 + t \cdot \left(t \cdot \left(0.5 + -0.16666666666666666 \cdot t\right) - 1\right)}\right)}^{c\_p}} \]
   12. Step-by-step derivation

       1. *-commutative 97.4%

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

   13. Simplified 97.4%

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

   if 6.20000000000000018 < c_p

   1. Initial program 12.5%

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

      1. associate-/l/ 12.5%

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

   3. Simplified 12.5%

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

   5. Taylor expanded in c_n around 0 12.5%

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

   6. Taylor expanded in t around 0 12.5%

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

   7. Taylor expanded in c_p around 0 2.1%

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

   8. Taylor expanded in t around inf 75.8%

      \[\leadsto \frac{1}{{\color{blue}{\left(\frac{-6}{{t}^{3}}\right)}}^{c\_p}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 6.2:\\ \;\;\;\;\frac{{\left(2 + s \cdot \left(1 + s \cdot \left(0.5 + s \cdot 0.16666666666666666\right)\right)\right)}^{\left(-c\_p\right)}}{{\left(\frac{1}{2 + t \cdot \left(-1 + t \cdot \left(0.5 + t \cdot -0.16666666666666666\right)\right)}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{-6}{{t}^{3}}\right)}^{c\_p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -750000000:\\ \;\;\;\;\frac{1}{{\left(\frac{-6}{{t}^{3}}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -750000000.0) (/ 1.0 (pow (/ -6.0 (pow t 3.0)) c_p)) 1.0))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -750000000.0) {
		tmp = 1.0 / pow((-6.0 / pow(t, 3.0)), c_p);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= (-750000000.0d0)) then
        tmp = 1.0d0 / (((-6.0d0) / (t ** 3.0d0)) ** c_p)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -750000000.0) {
		tmp = 1.0 / Math.pow((-6.0 / Math.pow(t, 3.0)), c_p);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -750000000.0:
		tmp = 1.0 / math.pow((-6.0 / math.pow(t, 3.0)), c_p)
	else:
		tmp = 1.0
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -750000000.0)
		tmp = Float64(1.0 / (Float64(-6.0 / (t ^ 3.0)) ^ c_p));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -750000000.0)
		tmp = 1.0 / ((-6.0 / (t ^ 3.0)) ^ c_p);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -750000000.0], N[(1.0 / N[Power[N[(-6.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if s < -7.5e8

   1. Initial program 16.7%

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

      1. associate-/l/ 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in c_n around 0 16.7%

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

   6. Taylor expanded in t around 0 16.7%

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

   7. Taylor expanded in c_p around 0 1.8%

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

   8. Taylor expanded in t around inf 100.0%

      \[\leadsto \frac{1}{{\color{blue}{\left(\frac{-6}{{t}^{3}}\right)}}^{c\_p}} \]

   if -7.5e8 < s

   1. Initial program 92.9%

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

      1. associate-/l/ 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in c_p around 0 96.4%

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

   6. Taylor expanded in c_n around 0 96.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.9% accurate, 835.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-/l/ 91.1%

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

3. Simplified 91.1%

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

5. Taylor expanded in c_p around 0 94.3%

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

6. Taylor expanded in c_n around 0 93.9%

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

Developer Target 1: 96.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024177 
(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))))