Harley's example

Percentage Accurate: 91.0% → 95.9%

Time: 1.2min

Alternatives: 7

Speedup: 835.0×

• could not determine a ground truth

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

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+113} \lor \neg \left(-s \leq 2 \cdot 10^{-100}\right):\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{-t}}\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (or (<= (- s) -2e+113) (not (<= (- s) 2e-100)))
   (pow
    (/ 1.0 (+ 2.0 (* s (+ (* s (+ 0.5 (* s -0.16666666666666666))) -1.0))))
    c_p)
   (/ (pow 0.5 c_n) (pow (+ 1.0 (/ -1.0 (+ 1.0 (exp (- t))))) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if ((-s <= -2e+113) || !(-s <= 2e-100)) {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	} else {
		tmp = pow(0.5, c_n) / pow((1.0 + (-1.0 / (1.0 + exp(-t)))), c_n);
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((-s <= (-2d+113)) .or. (.not. (-s <= 2d-100))) then
        tmp = (1.0d0 / (2.0d0 + (s * ((s * (0.5d0 + (s * (-0.16666666666666666d0)))) + (-1.0d0))))) ** c_p
    else
        tmp = (0.5d0 ** c_n) / ((1.0d0 + ((-1.0d0) / (1.0d0 + exp(-t)))) ** c_n)
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if ((-s <= -2e+113) || !(-s <= 2e-100)) {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	} else {
		tmp = Math.pow(0.5, c_n) / Math.pow((1.0 + (-1.0 / (1.0 + Math.exp(-t)))), c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if (-s <= -2e+113) or not (-s <= 2e-100):
		tmp = math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p)
	else:
		tmp = math.pow(0.5, c_n) / math.pow((1.0 + (-1.0 / (1.0 + math.exp(-t)))), c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if ((Float64(-s) <= -2e+113) || !(Float64(-s) <= 2e-100))
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	else
		tmp = Float64((0.5 ^ c_n) / (Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(Float64(-t))))) ^ c_n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if ((-s <= -2e+113) || ~((-s <= 2e-100)))
		tmp = (1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	else
		tmp = (0.5 ^ c_n) / ((1.0 + (-1.0 / (1.0 + exp(-t)))) ^ c_n);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[Or[LessEqual[(-s), -2e+113], N[Not[LessEqual[(-s), 2e-100]], $MachinePrecision]], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(s * N[(0.5 + N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < -2e113 or 2e-100 < (neg.f64 s)

   1. Initial program 81.8%

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

      1. associate-/l/ 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in c_n around 0 80.1%

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

   6. Taylor expanded in c_p around 0 86.2%

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

   7. Taylor expanded in s around 0 100.0%

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

   if -2e113 < (neg.f64 s) < 2e-100

   1. Initial program 95.2%

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

      1. associate-/l/ 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in s around 0 96.1%

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

   6. Taylor expanded in c_p around 0 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+113} \lor \neg \left(-s \leq 2 \cdot 10^{-100}\right):\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{-t}}\right)}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-100}:\\ \;\;\;\;e^{\left(c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 2e-100)
   (exp
    (+
     (-
      (*
       c_n
       (- (log1p (/ 1.0 (+ (exp s) 1.0))) (log1p (/ 1.0 (+ 1.0 (exp t))))))
      (* c_p (log1p (exp s))))
     (* c_p (log1p (exp t)))))
   (pow
    (/ 1.0 (+ 2.0 (* s (+ (* s (+ 0.5 (* s -0.16666666666666666))) -1.0))))
    c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 2e-100) {
		tmp = exp((((c_n * (log1p((1.0 / (exp(s) + 1.0))) - log1p((1.0 / (1.0 + exp(t)))))) - (c_p * log1p(exp(s)))) + (c_p * log1p(exp(t)))));
	} else {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 2e-100) {
		tmp = Math.exp((((c_n * (Math.log1p((1.0 / (Math.exp(s) + 1.0))) - Math.log1p((1.0 / (1.0 + Math.exp(t)))))) - (c_p * Math.log1p(Math.exp(s)))) + (c_p * Math.log1p(Math.exp(t)))));
	} else {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 2e-100:
		tmp = math.exp((((c_n * (math.log1p((1.0 / (math.exp(s) + 1.0))) - math.log1p((1.0 / (1.0 + math.exp(t)))))) - (c_p * math.log1p(math.exp(s)))) + (c_p * math.log1p(math.exp(t)))))
	else:
		tmp = math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 2e-100)
		tmp = exp(Float64(Float64(Float64(c_n * Float64(log1p(Float64(1.0 / Float64(exp(s) + 1.0))) - log1p(Float64(1.0 / Float64(1.0 + exp(t)))))) - Float64(c_p * log1p(exp(s)))) + Float64(c_p * log1p(exp(t)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 2e-100], N[Exp[N[(N[(N[(c$95$n * N[(N[Log[1 + N[(1.0 / N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(1.0 / N[(1.0 + N[Exp[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c$95$p * N[Log[1 + N[Exp[s], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * N[Log[1 + N[Exp[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(s * N[(0.5 + N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 2e-100

   1. Initial program 93.4%

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

      1. associate-/l/ 93.4%

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

   3. Simplified 93.4%

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

   6. Applied egg-rr 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate--l+ 98.7%

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

      3. distribute-lft-out-- 98.7%

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

   8. Simplified 98.7%

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

   if 2e-100 < (neg.f64 s)

   1. Initial program 88.5%

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

      1. associate-/l/ 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in c_n around 0 90.8%

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

   6. Taylor expanded in c_p around 0 97.7%

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

   7. Taylor expanded in s around 0 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-100}:\\ \;\;\;\;e^{\left(c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-t \leq 4 \cdot 10^{-42}:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- t) 4e-42)
   (/
    (pow
     (/ 1.0 (+ 2.0 (* s (+ (* s (+ 0.5 (* s -0.16666666666666666))) -1.0))))
     c_p)
    (pow (/ 1.0 (+ 1.0 (exp (- t)))) c_p))
   (/ (pow 0.5 c_n) (pow (+ 0.5 (* t -0.25)) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= 4e-42) {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p) / pow((1.0 / (1.0 + exp(-t))), c_p);
	} else {
		tmp = pow(0.5, c_n) / pow((0.5 + (t * -0.25)), c_n);
	}
	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 (-t <= 4d-42) then
        tmp = ((1.0d0 / (2.0d0 + (s * ((s * (0.5d0 + (s * (-0.16666666666666666d0)))) + (-1.0d0))))) ** c_p) / ((1.0d0 / (1.0d0 + exp(-t))) ** c_p)
    else
        tmp = (0.5d0 ** c_n) / ((0.5d0 + (t * (-0.25d0))) ** c_n)
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= 4e-42) {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p) / Math.pow((1.0 / (1.0 + Math.exp(-t))), c_p);
	} else {
		tmp = Math.pow(0.5, c_n) / Math.pow((0.5 + (t * -0.25)), c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -t <= 4e-42:
		tmp = math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p) / math.pow((1.0 / (1.0 + math.exp(-t))), c_p)
	else:
		tmp = math.pow(0.5, c_n) / math.pow((0.5 + (t * -0.25)), c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-t) <= 4e-42)
		tmp = Float64((Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))) + -1.0)))) ^ c_p) / (Float64(1.0 / Float64(1.0 + exp(Float64(-t)))) ^ c_p));
	else
		tmp = Float64((0.5 ^ c_n) / (Float64(0.5 + Float64(t * -0.25)) ^ c_n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-t <= 4e-42)
		tmp = ((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))) ^ c_p) / ((1.0 / (1.0 + exp(-t))) ^ c_p);
	else
		tmp = (0.5 ^ c_n) / ((0.5 + (t * -0.25)) ^ c_n);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-t), 4e-42], N[(N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(s * N[(0.5 + N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] / N[Power[N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(0.5 + N[(t * -0.25), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 4.00000000000000015e-42

   1. Initial program 94.8%

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

      1. associate-/l/ 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in c_n around 0 95.4%

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

   6. Taylor expanded in s around 0 97.5%

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

   if 4.00000000000000015e-42 < (neg.f64 t)

   1. Initial program 71.4%

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

      1. associate-/l/ 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in s around 0 75.4%

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

   6. Taylor expanded in t around 0 75.0%

      \[\leadsto \frac{\frac{{0.5}^{c\_n} \cdot {0.5}^{c\_p}}{{\color{blue}{\left(0.5 + t \cdot \left(0.020833333333333332 \cdot {t}^{2} - 0.25\right)\right)}}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]

   7. Taylor expanded in c_p around 0 87.5%

      \[\leadsto \color{blue}{\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot \left(0.020833333333333332 \cdot {t}^{2} - 0.25\right)\right)}^{c\_n}}} \]

   8. Taylor expanded in t around 0 100.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(0.5 + \color{blue}{-0.25 \cdot t}\right)}^{c\_n}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{{0.5}^{c\_n}}{{\left(0.5 + \color{blue}{t \cdot -0.25}\right)}^{c\_n}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(0.5 + \color{blue}{t \cdot -0.25}\right)}^{c\_n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 4 \cdot 10^{-42}:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+113} \lor \neg \left(-s \leq 2 \cdot 10^{-100}\right):\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;1 + t \cdot \left(c\_n \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (or (<= (- s) -2e+113) (not (<= (- s) 2e-100)))
   (pow
    (/ 1.0 (+ 2.0 (* s (+ (* s (+ 0.5 (* s -0.16666666666666666))) -1.0))))
    c_p)
   (+ 1.0 (* t (* c_n 0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if ((-s <= -2e+113) || !(-s <= 2e-100)) {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	} else {
		tmp = 1.0 + (t * (c_n * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((-s <= (-2d+113)) .or. (.not. (-s <= 2d-100))) then
        tmp = (1.0d0 / (2.0d0 + (s * ((s * (0.5d0 + (s * (-0.16666666666666666d0)))) + (-1.0d0))))) ** c_p
    else
        tmp = 1.0d0 + (t * (c_n * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if ((-s <= -2e+113) || !(-s <= 2e-100)) {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	} else {
		tmp = 1.0 + (t * (c_n * 0.5));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if (-s <= -2e+113) or not (-s <= 2e-100):
		tmp = math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p)
	else:
		tmp = 1.0 + (t * (c_n * 0.5))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if ((Float64(-s) <= -2e+113) || !(Float64(-s) <= 2e-100))
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	else
		tmp = Float64(1.0 + Float64(t * Float64(c_n * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if ((-s <= -2e+113) || ~((-s <= 2e-100)))
		tmp = (1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	else
		tmp = 1.0 + (t * (c_n * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[Or[LessEqual[(-s), -2e+113], N[Not[LessEqual[(-s), 2e-100]], $MachinePrecision]], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(s * N[(0.5 + N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[(1.0 + N[(t * N[(c$95$n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < -2e113 or 2e-100 < (neg.f64 s)

   1. Initial program 81.8%

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

      1. associate-/l/ 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in c_n around 0 80.1%

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

   6. Taylor expanded in c_p around 0 86.2%

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

   7. Taylor expanded in s around 0 100.0%

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

   if -2e113 < (neg.f64 s) < 2e-100

   1. Initial program 95.2%

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

      1. associate-/l/ 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in s around 0 96.1%

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

   6. Taylor expanded in t around 0 96.1%

      \[\leadsto \frac{\frac{{0.5}^{c\_n} \cdot {0.5}^{c\_p}}{{\color{blue}{\left(0.5 + t \cdot \left(0.020833333333333332 \cdot {t}^{2} - 0.25\right)\right)}}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]

   7. Taylor expanded in c_p around 0 98.0%

      \[\leadsto \color{blue}{\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot \left(0.020833333333333332 \cdot {t}^{2} - 0.25\right)\right)}^{c\_n}}} \]

   8. Taylor expanded in t around 0 98.7%

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

      1. associate-*r* 98.7%

         \[\leadsto 1 + \color{blue}{\left(0.5 \cdot c\_n\right) \cdot t} \]

   10. Simplified 98.7%

       \[\leadsto \color{blue}{1 + \left(0.5 \cdot c\_n\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+113} \lor \neg \left(-s \leq 2 \cdot 10^{-100}\right):\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;1 + t \cdot \left(c\_n \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{2 + s \cdot \left(s \cdot 0.5 + -1\right)}\right)}^{c\_p} \end{array} \]

FPCore

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

C

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

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 / (2.0d0 + (s * ((s * 0.5d0) + (-1.0d0))))) ** c_p
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p);
}

Python

def code(c_p, c_n, t, s):
	return math.pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p)

Julia

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

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = (1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))) ^ c_p;
end

Wolfram

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

Derivation

1. Initial program 92.6%

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

   1. associate-/l/ 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in c_n around 0 93.6%

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

6. Taylor expanded in c_p around 0 93.7%

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

7. Taylor expanded in s around 0 94.8%

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

8. Final simplification 94.8%

   \[\leadsto {\left(\frac{1}{2 + s \cdot \left(s \cdot 0.5 + -1\right)}\right)}^{c\_p} \]
9. Add Preprocessing

Alternative 6: 94.3% accurate, 119.3× speedup

Math

\[\begin{array}{l} \\ 1 + t \cdot \left(c\_n \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s) :precision binary64 (+ 1.0 (* t (* c_n 0.5))))

C

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

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = 1.0d0 + (t * (c_n * 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 + N[(t * N[(c$95$n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.6%

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

   1. associate-/l/ 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in s around 0 91.5%

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

6. Taylor expanded in t around 0 91.5%

   \[\leadsto \frac{\frac{{0.5}^{c\_n} \cdot {0.5}^{c\_p}}{{\color{blue}{\left(0.5 + t \cdot \left(0.020833333333333332 \cdot {t}^{2} - 0.25\right)\right)}}^{c\_n}}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}} \]

7. Taylor expanded in c_p around 0 93.4%

   \[\leadsto \color{blue}{\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot \left(0.020833333333333332 \cdot {t}^{2} - 0.25\right)\right)}^{c\_n}}} \]

8. Taylor expanded in t around 0 94.4%

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

   1. associate-*r* 94.4%

      \[\leadsto 1 + \color{blue}{\left(0.5 \cdot c\_n\right) \cdot t} \]

10. Simplified 94.4%

    \[\leadsto \color{blue}{1 + \left(0.5 \cdot c\_n\right) \cdot t} \]

11. Final simplification 94.4%

    \[\leadsto 1 + t \cdot \left(c\_n \cdot 0.5\right) \]
12. Add Preprocessing

Alternative 7: 94.3% 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 92.6%

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

   1. associate-/l/ 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in c_n around 0 93.6%

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

6. Taylor expanded in c_p around 0 94.4%

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

7. Final simplification 94.4%

   \[\leadsto 1 \]
8. Add Preprocessing

Developer target: 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 2024073 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

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

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