Harley's example

Percentage Accurate: 91.4% → 98.1%

Time: 2.4min

Alternatives: 8

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: 91.4% 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: 98.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{{\left(1 + e^{s}\right)}^{\left(-c\_p\right)}}{{0.5}^{c\_p}}\\ \mathbf{elif}\;-s \leq 100000000:\\ \;\;\;\;1 + s \cdot \left(c\_p \cdot 0.5 - s \cdot \left(c\_p \cdot \left(0.125 - c\_p \cdot 0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) -5e+22)
   (/ (pow (+ 1.0 (exp s)) (- c_p)) (pow 0.5 c_p))
   (if (<= (- s) 100000000.0)
     (+ 1.0 (* s (- (* c_p 0.5) (* s (* c_p (- 0.125 (* c_p 0.125)))))))
     (/ (pow (- (exp s)) c_n) (pow 0.5 c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= -5e+22) {
		tmp = pow((1.0 + exp(s)), -c_p) / pow(0.5, c_p);
	} else if (-s <= 100000000.0) {
		tmp = 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))));
	} else {
		tmp = pow(-exp(s), c_n) / pow(0.5, 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 <= (-5d+22)) then
        tmp = ((1.0d0 + exp(s)) ** -c_p) / (0.5d0 ** c_p)
    else if (-s <= 100000000.0d0) then
        tmp = 1.0d0 + (s * ((c_p * 0.5d0) - (s * (c_p * (0.125d0 - (c_p * 0.125d0))))))
    else
        tmp = (-exp(s) ** c_n) / (0.5d0 ** 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 <= -5e+22) {
		tmp = Math.pow((1.0 + Math.exp(s)), -c_p) / Math.pow(0.5, c_p);
	} else if (-s <= 100000000.0) {
		tmp = 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))));
	} else {
		tmp = Math.pow(-Math.exp(s), c_n) / Math.pow(0.5, c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= -5e+22:
		tmp = math.pow((1.0 + math.exp(s)), -c_p) / math.pow(0.5, c_p)
	elif -s <= 100000000.0:
		tmp = 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))))
	else:
		tmp = math.pow(-math.exp(s), c_n) / math.pow(0.5, c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= -5e+22)
		tmp = Float64((Float64(1.0 + exp(s)) ^ Float64(-c_p)) / (0.5 ^ c_p));
	elseif (Float64(-s) <= 100000000.0)
		tmp = Float64(1.0 + Float64(s * Float64(Float64(c_p * 0.5) - Float64(s * Float64(c_p * Float64(0.125 - Float64(c_p * 0.125)))))));
	else
		tmp = Float64((Float64(-exp(s)) ^ c_n) / (0.5 ^ c_n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= -5e+22)
		tmp = ((1.0 + exp(s)) ^ -c_p) / (0.5 ^ c_p);
	elseif (-s <= 100000000.0)
		tmp = 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))));
	else
		tmp = (-exp(s) ^ c_n) / (0.5 ^ c_n);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), -5e+22], 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), 100000000.0], N[(1.0 + N[(s * N[(N[(c$95$p * 0.5), $MachinePrecision] - N[(s * N[(c$95$p * N[(0.125 - N[(c$95$p * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[(-N[Exp[s], $MachinePrecision]), c$95$n], $MachinePrecision] / N[Power[0.5, c$95$n], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (neg.f64 s) < -4.9999999999999996e22

   1. Initial program 42.9%

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

      1. associate-/l/ 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in c_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%

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

      1. *-un-lft-identity 3.1%

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

      2. inv-pow 3.1%

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

      3. pow-pow 3.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 100.0%

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

      6. sqr-neg 100.0%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -4.9999999999999996e22 < (neg.f64 s) < 1e8

   1. Initial program 93.0%

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

      1. associate-/l/ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in c_n around 0 94.5%

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

   6. Taylor expanded in t around 0 95.7%

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

   7. Taylor expanded in s around 0 98.4%

      \[\leadsto \color{blue}{1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_p + 0.125 \cdot {c\_p}^{2}\right)\right)} \]

   8. Taylor expanded in c_p around 0 98.4%

      \[\leadsto 1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \color{blue}{\left(c\_p \cdot \left(0.125 \cdot c\_p - 0.125\right)\right)}\right) \]

   if 1e8 < (neg.f64 s)

   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 c_p around 0 3.1%

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

   6. Taylor expanded in t around 0 3.1%

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

      1. *-un-lft-identity 3.1%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. associate--r+ 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{{\left(\color{blue}{0} - e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}} \]

      5. neg-sub0 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\frac{{\left(1 + e^{s}\right)}^{\left(-c\_p\right)}}{{0.5}^{c\_p}}\\ \mathbf{elif}\;-s \leq 100000000:\\ \;\;\;\;1 + s \cdot \left(c\_p \cdot 0.5 - s \cdot \left(c\_p \cdot \left(0.125 - c\_p \cdot 0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 2e-40:
		tmp = math.pow((1.0 + math.exp(s)), -c_p) / math.pow(0.5, c_p)
	else:
		tmp = math.exp(((c_p * math.log1p((1.0 - t))) - (c_p * math.log1p(math.exp(-s)))))
	return tmp

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if c_p < 1.9999999999999999e-40

   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_n around 0 93.3%

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

   6. Taylor expanded in t around 0 94.7%

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

      1. *-un-lft-identity 94.7%

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

      2. inv-pow 94.7%

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

      3. pow-pow 94.7%

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

      4. add-sqr-sqrt 45.5%

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

      5. sqrt-unprod 97.5%

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

      6. sqr-neg 97.5%

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

      7. sqrt-unprod 51.9%

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

      8. add-sqr-sqrt 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. *-lft-identity 97.9%

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

      2. neg-mul-1 97.9%

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

   10. Simplified 97.9%

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

   if 1.9999999999999999e-40 < c_p

   1. Initial program 82.9%

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

      1. associate-/l/ 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in c_n around 0 82.9%

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

      1. add-exp-log 82.9%

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

      2. log-div 82.9%

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

      3. log-pow 83.3%

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

      4. log-rec 83.3%

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

      5. log1p-define 83.3%

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

      6. log-pow 100.0%

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

      7. neg-log 100.0%

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

      8. distribute-rgt-neg-out 100.0%

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

      9. log1p-define 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in t around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 98.3%

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

Alternative 3: 96.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -740000000:\\ \;\;\;\;\frac{{\left(-e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;1 + s \cdot \left(c\_p \cdot 0.5 - s \cdot \left(c\_p \cdot \left(0.125 - c\_p \cdot 0.125\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -740000000.0)
   (/ (pow (- (exp s)) c_n) (pow 0.5 c_n))
   (+ 1.0 (* s (- (* c_p 0.5) (* s (* c_p (- 0.125 (* c_p 0.125)))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -740000000.0) {
		tmp = pow(-exp(s), c_n) / pow(0.5, c_n);
	} else {
		tmp = 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))));
	}
	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 <= (-740000000.0d0)) then
        tmp = (-exp(s) ** c_n) / (0.5d0 ** c_n)
    else
        tmp = 1.0d0 + (s * ((c_p * 0.5d0) - (s * (c_p * (0.125d0 - (c_p * 0.125d0))))))
    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 <= -740000000.0) {
		tmp = Math.pow(-Math.exp(s), c_n) / Math.pow(0.5, c_n);
	} else {
		tmp = 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -740000000.0:
		tmp = math.pow(-math.exp(s), c_n) / math.pow(0.5, c_n)
	else:
		tmp = 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -740000000.0)
		tmp = Float64((Float64(-exp(s)) ^ c_n) / (0.5 ^ c_n));
	else
		tmp = Float64(1.0 + Float64(s * Float64(Float64(c_p * 0.5) - Float64(s * Float64(c_p * Float64(0.125 - Float64(c_p * 0.125)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -740000000.0)
		tmp = (-exp(s) ^ c_n) / (0.5 ^ c_n);
	else
		tmp = 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -740000000.0], N[(N[Power[(-N[Exp[s], $MachinePrecision]), c$95$n], $MachinePrecision] / N[Power[0.5, c$95$n], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(s * N[(N[(c$95$p * 0.5), $MachinePrecision] - N[(s * N[(c$95$p * N[(0.125 - N[(c$95$p * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -7.4e8

   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 c_p around 0 3.1%

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

   6. Taylor expanded in t around 0 3.1%

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

      1. *-un-lft-identity 3.1%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. associate--r+ 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \frac{{\left(\color{blue}{0} - e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}} \]

      5. neg-sub0 100.0%

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

   10. Simplified 100.0%

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

   if -7.4e8 < s

   1. Initial program 91.6%

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

      1. associate-/l/ 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in c_n around 0 91.9%

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

   6. Taylor expanded in t around 0 93.1%

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

   7. Taylor expanded in s around 0 95.7%

      \[\leadsto \color{blue}{1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_p + 0.125 \cdot {c\_p}^{2}\right)\right)} \]

   8. Taylor expanded in c_p around 0 95.7%

      \[\leadsto 1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \color{blue}{\left(c\_p \cdot \left(0.125 \cdot c\_p - 0.125\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq -740000000:\\ \;\;\;\;\frac{{\left(-e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;1 + s \cdot \left(c\_p \cdot 0.5 - s \cdot \left(c\_p \cdot \left(0.125 - c\_p \cdot 0.125\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 29:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot 0.5 + -1\right)}\right)}^{c\_p}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;1 + t \cdot \left(0.5 \cdot c\_n\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 29.0)
   (/ (pow (/ 1.0 (+ 2.0 (* s (+ (* s 0.5) -1.0)))) c_p) (pow 0.5 c_p))
   (+ 1.0 (* t (* 0.5 c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 29.0) {
		tmp = pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p) / pow(0.5, c_p);
	} else {
		tmp = 1.0 + (t * (0.5 * 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 (c_p <= 29.0d0) then
        tmp = ((1.0d0 / (2.0d0 + (s * ((s * 0.5d0) + (-1.0d0))))) ** c_p) / (0.5d0 ** c_p)
    else
        tmp = 1.0d0 + (t * (0.5d0 * 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 (c_p <= 29.0) {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p) / Math.pow(0.5, c_p);
	} else {
		tmp = 1.0 + (t * (0.5 * c_n));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 29.0:
		tmp = math.pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p) / math.pow(0.5, c_p)
	else:
		tmp = 1.0 + (t * (0.5 * c_n))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 29.0)
		tmp = Float64((Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * 0.5) + -1.0)))) ^ c_p) / (0.5 ^ c_p));
	else
		tmp = Float64(1.0 + Float64(t * Float64(0.5 * c_n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_p <= 29.0)
		tmp = ((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))) ^ c_p) / (0.5 ^ c_p);
	else
		tmp = 1.0 + (t * (0.5 * c_n));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 29.0], N[(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] / N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t * N[(0.5 * c$95$n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 29

   1. Initial program 94.0%

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

      1. associate-/l/ 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in c_n around 0 94.3%

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

   6. Taylor expanded in t around 0 95.5%

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

   7. Taylor expanded in s around 0 96.1%

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

   if 29 < c_p

   1. Initial program 11.1%

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

      1. associate-/l/ 11.1%

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

   3. Simplified 11.1%

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

   5. Taylor expanded in c_p around 0 78.5%

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

   6. Taylor expanded in s around 0 78.5%

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

   7. Taylor expanded in t around 0 78.5%

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

      1. associate-*r* 78.5%

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

   9. Simplified 78.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 29:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot 0.5 + -1\right)}\right)}^{c\_p}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;1 + t \cdot \left(0.5 \cdot c\_n\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 49.1× speedup

Math

\[\begin{array}{l} \\ 1 + s \cdot \left(c\_p \cdot 0.5 - s \cdot \left(c\_p \cdot \left(0.125 - c\_p \cdot 0.125\right)\right)\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (+ 1.0 (* s (- (* c_p 0.5) (* s (* c_p (- 0.125 (* c_p 0.125))))))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))));
}

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 + (s * ((c_p * 0.5d0) - (s * (c_p * (0.125d0 - (c_p * 0.125d0))))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 + Float64(s * Float64(Float64(c_p * 0.5) - Float64(s * Float64(c_p * Float64(0.125 - Float64(c_p * 0.125)))))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + (s * ((c_p * 0.5) - (s * (c_p * (0.125 - (c_p * 0.125))))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 + N[(s * N[(N[(c$95$p * 0.5), $MachinePrecision] - N[(s * N[(c$95$p * N[(0.125 - N[(c$95$p * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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_n around 0 91.4%

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

6. Taylor expanded in t around 0 92.5%

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

7. Taylor expanded in s around 0 93.2%

   \[\leadsto \color{blue}{1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_p + 0.125 \cdot {c\_p}^{2}\right)\right)} \]

8. Taylor expanded in c_p around 0 93.2%

   \[\leadsto 1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \color{blue}{\left(c\_p \cdot \left(0.125 \cdot c\_p - 0.125\right)\right)}\right) \]

9. Final simplification 93.2%

   \[\leadsto 1 + s \cdot \left(c\_p \cdot 0.5 - s \cdot \left(c\_p \cdot \left(0.125 - c\_p \cdot 0.125\right)\right)\right) \]
10. Add Preprocessing

Alternative 6: 94.7% accurate, 75.9× speedup

Math

\[\begin{array}{l} \\ 1 + \left(c\_p \cdot s\right) \cdot \left(0.5 + s \cdot -0.125\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (+ 1.0 (* (* c_p s) (+ 0.5 (* s -0.125)))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + ((c_p * s) * (0.5 + (s * -0.125)));
}

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 + ((c_p * s) * (0.5d0 + (s * (-0.125d0))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 + ((c_p * s) * (0.5 + (s * -0.125)));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + ((c_p * s) * (0.5 + (s * -0.125)))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 + Float64(Float64(c_p * s) * Float64(0.5 + Float64(s * -0.125))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + ((c_p * s) * (0.5 + (s * -0.125)));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 + N[(N[(c$95$p * s), $MachinePrecision] * N[(0.5 + N[(s * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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_n around 0 91.4%

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

6. Taylor expanded in t around 0 92.5%

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

7. Taylor expanded in s around 0 93.2%

   \[\leadsto \color{blue}{1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_p + 0.125 \cdot {c\_p}^{2}\right)\right)} \]

8. Taylor expanded in c_p around 0 93.1%

   \[\leadsto 1 + \color{blue}{c\_p \cdot \left(s \cdot \left(0.5 + -0.125 \cdot s\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 93.1%

      \[\leadsto 1 + \color{blue}{\left(c\_p \cdot s\right) \cdot \left(0.5 + -0.125 \cdot s\right)} \]

   2. *-commutative 93.1%

      \[\leadsto 1 + \left(c\_p \cdot s\right) \cdot \left(0.5 + \color{blue}{s \cdot -0.125}\right) \]

10. Simplified 93.1%

    \[\leadsto 1 + \color{blue}{\left(c\_p \cdot s\right) \cdot \left(0.5 + s \cdot -0.125\right)} \]
11. Add Preprocessing

Alternative 7: 94.7% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

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 * (c_p * s))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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_n around 0 91.4%

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

6. Taylor expanded in t around 0 92.5%

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

7. Taylor expanded in s around 0 93.1%

   \[\leadsto \color{blue}{1 + 0.5 \cdot \left(c\_p \cdot s\right)} \]
8. Add Preprocessing

Alternative 8: 94.7% 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 93.0%

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

Developer Target 1: 96.7% 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 2024150 
(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))))