Harley's example

Percentage Accurate: 90.6% → 95.6%

Time: 1.4min

Alternatives: 6

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: 90.6% 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: 95.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{{\left(1 + \frac{-1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 2e-53)
   (/
    (pow (+ 1.0 (/ -1.0 (+ 1.0 (exp (- s))))) c_n)
    (pow (+ 1.0 (/ 1.0 (- -1.0 (exp (- t))))) c_n))
   (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-53) {
		tmp = pow((1.0 + (-1.0 / (1.0 + exp(-s)))), c_n) / pow((1.0 + (1.0 / (-1.0 - exp(-t)))), c_n);
	} else {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), 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 <= 2d-53) then
        tmp = ((1.0d0 + ((-1.0d0) / (1.0d0 + exp(-s)))) ** c_n) / ((1.0d0 + (1.0d0 / ((-1.0d0) - exp(-t)))) ** c_n)
    else
        tmp = (1.0d0 / (2.0d0 + (s * ((s * (0.5d0 + (s * (-0.16666666666666666d0)))) + (-1.0d0))))) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 2e-53) {
		tmp = Math.pow((1.0 + (-1.0 / (1.0 + Math.exp(-s)))), c_n) / Math.pow((1.0 + (1.0 / (-1.0 - Math.exp(-t)))), c_n);
	} 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-53:
		tmp = math.pow((1.0 + (-1.0 / (1.0 + math.exp(-s)))), c_n) / math.pow((1.0 + (1.0 / (-1.0 - math.exp(-t)))), c_n)
	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-53)
		tmp = Float64((Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(Float64(-s))))) ^ c_n) / (Float64(1.0 + Float64(1.0 / Float64(-1.0 - exp(Float64(-t))))) ^ c_n));
	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

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 2e-53)
		tmp = ((1.0 + (-1.0 / (1.0 + exp(-s)))) ^ c_n) / ((1.0 + (1.0 / (-1.0 - exp(-t)))) ^ c_n);
	else
		tmp = (1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 2e-53], N[(N[Power[N[(1.0 + N[(-1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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], 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) < 2.00000000000000006e-53

   1. Initial program 95.7%

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

      1. associate-/l/ 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in c_p around 0 99.1%

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

   if 2.00000000000000006e-53 < (neg.f64 s)

   1. Initial program 82.7%

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

      1. associate-/l/ 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in c_p around 0 91.4%

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

   6. Taylor expanded in c_n around 0 95.8%

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

   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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{{\left(1 + \frac{-1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 2e-53)
   (/ (pow 0.5 c_n) (pow (+ 1.0 (/ 1.0 (- -1.0 (exp (- t))))) c_n))
   (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-53) {
		tmp = pow(0.5, c_n) / pow((1.0 + (1.0 / (-1.0 - exp(-t)))), c_n);
	} else {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), 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 <= 2d-53) then
        tmp = (0.5d0 ** c_n) / ((1.0d0 + (1.0d0 / ((-1.0d0) - exp(-t)))) ** c_n)
    else
        tmp = (1.0d0 / (2.0d0 + (s * ((s * (0.5d0 + (s * (-0.16666666666666666d0)))) + (-1.0d0))))) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 2e-53) {
		tmp = Math.pow(0.5, c_n) / Math.pow((1.0 + (1.0 / (-1.0 - Math.exp(-t)))), c_n);
	} 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-53:
		tmp = math.pow(0.5, c_n) / math.pow((1.0 + (1.0 / (-1.0 - math.exp(-t)))), c_n)
	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-53)
		tmp = Float64((0.5 ^ c_n) / (Float64(1.0 + Float64(1.0 / Float64(-1.0 - exp(Float64(-t))))) ^ c_n));
	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

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 2e-53)
		tmp = (0.5 ^ c_n) / ((1.0 + (1.0 / (-1.0 - exp(-t)))) ^ c_n);
	else
		tmp = (1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 2e-53], 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], 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) < 2.00000000000000006e-53

   1. Initial program 95.7%

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

      1. associate-/l/ 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in c_p around 0 99.1%

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

   6. Taylor expanded in s around 0 98.7%

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

   if 2.00000000000000006e-53 < (neg.f64 s)

   1. Initial program 82.7%

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

      1. associate-/l/ 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in c_p around 0 91.4%

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

   6. Taylor expanded in c_n around 0 95.8%

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

   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 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 - \left(0.5 + t \cdot 0.25\right)\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 2e-53)
   (/ (pow 0.5 c_n) (pow (- 1.0 (+ 0.5 (* t 0.25))) c_n))
   (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-53) {
		tmp = pow(0.5, c_n) / pow((1.0 - (0.5 + (t * 0.25))), c_n);
	} else {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), 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 <= 2d-53) then
        tmp = (0.5d0 ** c_n) / ((1.0d0 - (0.5d0 + (t * 0.25d0))) ** c_n)
    else
        tmp = (1.0d0 / (2.0d0 + (s * ((s * (0.5d0 + (s * (-0.16666666666666666d0)))) + (-1.0d0))))) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 2e-53) {
		tmp = Math.pow(0.5, c_n) / Math.pow((1.0 - (0.5 + (t * 0.25))), c_n);
	} 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-53:
		tmp = math.pow(0.5, c_n) / math.pow((1.0 - (0.5 + (t * 0.25))), c_n)
	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-53)
		tmp = Float64((0.5 ^ c_n) / (Float64(1.0 - Float64(0.5 + Float64(t * 0.25))) ^ c_n));
	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

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 2e-53)
		tmp = (0.5 ^ c_n) / ((1.0 - (0.5 + (t * 0.25))) ^ c_n);
	else
		tmp = (1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 2e-53], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(1.0 - N[(0.5 + N[(t * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $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) < 2.00000000000000006e-53

   1. Initial program 95.7%

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

      1. associate-/l/ 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in c_p around 0 99.1%

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

   6. Taylor expanded in s around 0 98.7%

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

   7. Taylor expanded in t around 0 98.3%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(1 - \color{blue}{\left(0.5 + 0.25 \cdot t\right)}\right)}^{c\_n}} \]
   8. Step-by-step derivation

      1. *-commutative 98.3%

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

   9. Simplified 98.3%

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

   if 2.00000000000000006e-53 < (neg.f64 s)

   1. Initial program 82.7%

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

      1. associate-/l/ 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in c_p around 0 91.4%

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

   6. Taylor expanded in c_n around 0 95.8%

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

   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 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 - \left(0.5 + t \cdot 0.25\right)\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -1 \cdot 10^{-53}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(s \cdot c\_n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -1e-53)
   (pow
    (/ 1.0 (+ 2.0 (* s (+ (* s (+ 0.5 (* s -0.16666666666666666))) -1.0))))
    c_p)
   (+ 1.0 (* -0.5 (* s c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -1e-53) {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	} else {
		tmp = 1.0 + (-0.5 * (s * 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 <= (-1d-53)) then
        tmp = (1.0d0 / (2.0d0 + (s * ((s * (0.5d0 + (s * (-0.16666666666666666d0)))) + (-1.0d0))))) ** c_p
    else
        tmp = 1.0d0 + ((-0.5d0) * (s * 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 <= -1e-53) {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	} else {
		tmp = 1.0 + (-0.5 * (s * c_n));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -1e-53:
		tmp = math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p)
	else:
		tmp = 1.0 + (-0.5 * (s * c_n))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -1e-53)
		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(-0.5 * Float64(s * c_n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -1e-53)
		tmp = (1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	else
		tmp = 1.0 + (-0.5 * (s * c_n));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -1e-53], 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[(-0.5 * N[(s * c$95$n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -1.00000000000000003e-53

   1. Initial program 79.3%

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

      1. associate-/l/ 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in c_p around 0 91.8%

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

   6. Taylor expanded in c_n around 0 91.9%

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

   7. Taylor expanded in s around 0 96.0%

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

   if -1.00000000000000003e-53 < s

   1. Initial program 96.1%

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

      1. associate-/l/ 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in c_p around 0 99.1%

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

   6. Taylor expanded in t around 0 97.8%

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

   7. Taylor expanded in s around 0 98.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq -1 \cdot 10^{-53}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(s \cdot c\_n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 119.3× speedup

Math

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

FPCore

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

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (-0.5 * (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 + ((-0.5d0) * (s * c_n))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

   1. associate-/l/ 94.6%

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

3. Simplified 94.6%

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

5. Taylor expanded in c_p around 0 97.3%

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

6. Taylor expanded in t around 0 95.7%

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

7. Taylor expanded in s around 0 96.2%

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

8. Final simplification 96.2%

   \[\leadsto 1 + -0.5 \cdot \left(s \cdot c\_n\right) \]
9. Add Preprocessing

Alternative 6: 94.1% accurate, 835.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

   1. associate-/l/ 94.6%

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

3. Simplified 94.6%

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

5. Taylor expanded in c_p around 0 97.3%

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

6. Taylor expanded in c_n around 0 96.1%

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

Developer target: 96.1% 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 2024093 
(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))))