Harley's example

Percentage Accurate: 90.4% → 98.4%

Time: 1.2min

Alternatives: 6

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 2.7909604565087886e-282
  2. c_n = 7.44027303181204e-136
  3. t = -4.2536964727909385e+41
  4. s = -1.6253675792973146e-21

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: 90.4% 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: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 100000000:\\ \;\;\;\;\frac{1}{e^{\left(c\_p \cdot \mathsf{log1p}\left(e^{s}\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{t} + 1}\right) - \mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right)\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 100000000.0)
   (/
    1.0
    (exp
     (-
      (+
       (* c_p (log1p (exp s)))
       (*
        c_n
        (- (log1p (/ 1.0 (+ (exp t) 1.0))) (log1p (/ 1.0 (+ (exp s) 1.0))))))
      (* c_p (log1p (exp t))))))
   (/ (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 <= 100000000.0) {
		tmp = 1.0 / exp((((c_p * log1p(exp(s))) + (c_n * (log1p((1.0 / (exp(t) + 1.0))) - log1p((1.0 / (exp(s) + 1.0)))))) - (c_p * log1p(exp(t)))));
	} else {
		tmp = pow(-exp(s), c_n) / pow(0.5, c_n);
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 100000000.0) {
		tmp = 1.0 / Math.exp((((c_p * Math.log1p(Math.exp(s))) + (c_n * (Math.log1p((1.0 / (Math.exp(t) + 1.0))) - Math.log1p((1.0 / (Math.exp(s) + 1.0)))))) - (c_p * Math.log1p(Math.exp(t)))));
	} 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 <= 100000000.0:
		tmp = 1.0 / math.exp((((c_p * math.log1p(math.exp(s))) + (c_n * (math.log1p((1.0 / (math.exp(t) + 1.0))) - math.log1p((1.0 / (math.exp(s) + 1.0)))))) - (c_p * math.log1p(math.exp(t)))))
	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) <= 100000000.0)
		tmp = Float64(1.0 / exp(Float64(Float64(Float64(c_p * log1p(exp(s))) + Float64(c_n * Float64(log1p(Float64(1.0 / Float64(exp(t) + 1.0))) - log1p(Float64(1.0 / Float64(exp(s) + 1.0)))))) - Float64(c_p * log1p(exp(t))))));
	else
		tmp = Float64((Float64(-exp(s)) ^ c_n) / (0.5 ^ c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 100000000.0], N[(1.0 / N[Exp[N[(N[(N[(c$95$p * N[Log[1 + N[Exp[s], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(c$95$n * N[(N[Log[1 + N[(1.0 / N[(N[Exp[t], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(1.0 / N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c$95$p * N[Log[1 + N[Exp[t], $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 2 regimes

2. if (neg.f64 s) < 1e8

   1. Initial program 92.1%

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

      1. associate-/l/ 92.1%

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

   3. Simplified 92.1%

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

   6. Applied egg-rr 98.9%

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

      1. unpow-1 98.9%

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

      2. associate--l+ 99.0%

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

      3. distribute-lft-out-- 99.0%

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

   8. Simplified 99.0%

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

   if 1e8 < (neg.f64 s)

   1. Initial program 57.1%

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

      1. associate-/l/ 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in c_p around 0 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%

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

      1. *-lft-identity 100.0%

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

      2. associate--r+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. 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 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 100000000:\\ \;\;\;\;\frac{1}{e^{\left(c\_p \cdot \mathsf{log1p}\left(e^{s}\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{t} + 1}\right) - \mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right)\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 100000000:\\ \;\;\;\;e^{\left(c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - \mathsf{log1p}\left(\frac{1}{e^{t} + 1}\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 100000000.0)
   (exp
    (+
     (-
      (*
       c_n
       (- (log1p (/ 1.0 (+ (exp s) 1.0))) (log1p (/ 1.0 (+ (exp t) 1.0)))))
      (* c_p (log1p (exp s))))
     (* c_p (log1p (exp t)))))
   (/ (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 <= 100000000.0) {
		tmp = exp((((c_n * (log1p((1.0 / (exp(s) + 1.0))) - log1p((1.0 / (exp(t) + 1.0))))) - (c_p * log1p(exp(s)))) + (c_p * log1p(exp(t)))));
	} else {
		tmp = pow(-exp(s), c_n) / pow(0.5, c_n);
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 100000000.0) {
		tmp = Math.exp((((c_n * (Math.log1p((1.0 / (Math.exp(s) + 1.0))) - Math.log1p((1.0 / (Math.exp(t) + 1.0))))) - (c_p * Math.log1p(Math.exp(s)))) + (c_p * Math.log1p(Math.exp(t)))));
	} 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 <= 100000000.0:
		tmp = math.exp((((c_n * (math.log1p((1.0 / (math.exp(s) + 1.0))) - math.log1p((1.0 / (math.exp(t) + 1.0))))) - (c_p * math.log1p(math.exp(s)))) + (c_p * math.log1p(math.exp(t)))))
	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) <= 100000000.0)
		tmp = exp(Float64(Float64(Float64(c_n * Float64(log1p(Float64(1.0 / Float64(exp(s) + 1.0))) - log1p(Float64(1.0 / Float64(exp(t) + 1.0))))) - Float64(c_p * log1p(exp(s)))) + Float64(c_p * log1p(exp(t)))));
	else
		tmp = Float64((Float64(-exp(s)) ^ c_n) / (0.5 ^ c_n));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 1e8

   1. Initial program 92.1%

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

      1. associate-/l/ 92.1%

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

   3. Simplified 92.1%

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

   6. Applied egg-rr 98.9%

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

      1. *-lft-identity 98.9%

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

      2. associate--l+ 99.0%

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

      3. distribute-lft-out-- 99.0%

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

   8. Simplified 99.0%

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

   if 1e8 < (neg.f64 s)

   1. Initial program 57.1%

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

      1. associate-/l/ 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in c_p around 0 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%

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

      1. *-lft-identity 100.0%

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

      2. associate--r+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. 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 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 100000000:\\ \;\;\;\;e^{\left(c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - \mathsf{log1p}\left(\frac{1}{e^{t} + 1}\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 100000000.0)
   (/ (pow (+ (exp s) 1.0) (- c_p)) (pow (/ 1.0 (+ (exp (- t)) 1.0)) c_p))
   (/ (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 <= 100000000.0) {
		tmp = pow((exp(s) + 1.0), -c_p) / pow((1.0 / (exp(-t) + 1.0)), c_p);
	} 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 <= 100000000.0d0) then
        tmp = ((exp(s) + 1.0d0) ** -c_p) / ((1.0d0 / (exp(-t) + 1.0d0)) ** c_p)
    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 <= 100000000.0) {
		tmp = Math.pow((Math.exp(s) + 1.0), -c_p) / Math.pow((1.0 / (Math.exp(-t) + 1.0)), c_p);
	} 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 <= 100000000.0:
		tmp = math.pow((math.exp(s) + 1.0), -c_p) / math.pow((1.0 / (math.exp(-t) + 1.0)), c_p)
	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) <= 100000000.0)
		tmp = Float64((Float64(exp(s) + 1.0) ^ Float64(-c_p)) / (Float64(1.0 / Float64(exp(Float64(-t)) + 1.0)) ^ c_p));
	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 <= 100000000.0)
		tmp = ((exp(s) + 1.0) ^ -c_p) / ((1.0 / (exp(-t) + 1.0)) ^ c_p);
	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), 100000000.0], N[(N[Power[N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[N[(1.0 / N[(N[Exp[(-t)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], c$95$p], $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 2 regimes

2. if (neg.f64 s) < 1e8

   1. Initial program 92.1%

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

      1. associate-/l/ 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in c_n around 0 94.7%

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

      1. *-un-lft-identity 94.7%

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

      2. inv-pow 94.7%

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

      3. pow-pow 94.7%

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

      4. add-sqr-sqrt 47.0%

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

      5. sqrt-unprod 96.3%

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

      6. sqr-neg 96.3%

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

      7. sqrt-unprod 49.2%

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

      8. add-sqr-sqrt 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. *-lft-identity 97.4%

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

      2. neg-mul-1 97.4%

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

   9. Simplified 97.4%

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

   if 1e8 < (neg.f64 s)

   1. Initial program 57.1%

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

      1. associate-/l/ 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in c_p around 0 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%

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

      1. *-lft-identity 100.0%

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

      2. associate--r+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. 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 2 regimes into one program.

4. Final simplification 97.5%

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

Alternative 4: 95.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -750000000:\\ \;\;\;\;\frac{{\left(-e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{s} + 1\right)}^{\left(-c\_p\right)}}{{0.5}^{c\_p}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -750000000.0)
   (/ (pow (- (exp s)) c_n) (pow 0.5 c_n))
   (/ (pow (+ (exp s) 1.0) (- c_p)) (pow 0.5 c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -750000000.0) {
		tmp = pow(-exp(s), c_n) / pow(0.5, c_n);
	} else {
		tmp = pow((exp(s) + 1.0), -c_p) / pow(0.5, 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 <= (-750000000.0d0)) then
        tmp = (-exp(s) ** c_n) / (0.5d0 ** c_n)
    else
        tmp = ((exp(s) + 1.0d0) ** -c_p) / (0.5d0 ** 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 <= -750000000.0) {
		tmp = Math.pow(-Math.exp(s), c_n) / Math.pow(0.5, c_n);
	} else {
		tmp = Math.pow((Math.exp(s) + 1.0), -c_p) / Math.pow(0.5, c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -750000000.0:
		tmp = math.pow(-math.exp(s), c_n) / math.pow(0.5, c_n)
	else:
		tmp = math.pow((math.exp(s) + 1.0), -c_p) / math.pow(0.5, c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -750000000.0)
		tmp = Float64((Float64(-exp(s)) ^ c_n) / (0.5 ^ c_n));
	else
		tmp = Float64((Float64(exp(s) + 1.0) ^ Float64(-c_p)) / (0.5 ^ c_p));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -750000000.0)
		tmp = (-exp(s) ^ c_n) / (0.5 ^ c_n);
	else
		tmp = ((exp(s) + 1.0) ^ -c_p) / (0.5 ^ c_p);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -750000000.0], N[(N[Power[(-N[Exp[s], $MachinePrecision]), c$95$n], $MachinePrecision] / N[Power[0.5, c$95$n], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -7.5e8

   1. Initial program 57.1%

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

      1. associate-/l/ 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in c_p around 0 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%

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

      1. *-lft-identity 100.0%

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

      2. associate--r+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. 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 -7.5e8 < s

   1. Initial program 92.1%

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

      1. associate-/l/ 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in c_n around 0 94.7%

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

      1. *-un-lft-identity 94.7%

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

      2. inv-pow 94.7%

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

      3. pow-pow 94.7%

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

      4. add-sqr-sqrt 47.0%

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

      5. sqrt-unprod 96.3%

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

      6. sqr-neg 96.3%

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

      7. sqrt-unprod 49.2%

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

      8. add-sqr-sqrt 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. *-lft-identity 97.4%

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

      2. neg-mul-1 97.4%

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

   9. Simplified 97.4%

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

   10. Taylor expanded in t around 0 97.4%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq -750000000:\\ \;\;\;\;\frac{{\left(-e^{s}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{s} + 1\right)}^{\left(-c\_p\right)}}{{0.5}^{c\_p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 1100:\\ \;\;\;\;\frac{{\left(2 + s \cdot \left(s \cdot 0.5 + 1\right)\right)}^{\left(-c\_p\right)}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{t} + 1\right)}^{\left(-c\_p\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 1100.0)
   (/ (pow (+ 2.0 (* s (+ (* s 0.5) 1.0))) (- c_p)) (pow 0.5 c_p))
   (pow (+ (exp t) 1.0) (- c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 1100.0) {
		tmp = pow((2.0 + (s * ((s * 0.5) + 1.0))), -c_p) / pow(0.5, c_p);
	} else {
		tmp = pow((exp(t) + 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 (c_p <= 1100.0d0) then
        tmp = ((2.0d0 + (s * ((s * 0.5d0) + 1.0d0))) ** -c_p) / (0.5d0 ** c_p)
    else
        tmp = (exp(t) + 1.0d0) ** -c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 1100.0) {
		tmp = Math.pow((2.0 + (s * ((s * 0.5) + 1.0))), -c_p) / Math.pow(0.5, c_p);
	} else {
		tmp = Math.pow((Math.exp(t) + 1.0), -c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 1100.0:
		tmp = math.pow((2.0 + (s * ((s * 0.5) + 1.0))), -c_p) / math.pow(0.5, c_p)
	else:
		tmp = math.pow((math.exp(t) + 1.0), -c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 1100.0)
		tmp = Float64((Float64(2.0 + Float64(s * Float64(Float64(s * 0.5) + 1.0))) ^ Float64(-c_p)) / (0.5 ^ c_p));
	else
		tmp = Float64(exp(t) + 1.0) ^ Float64(-c_p);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_p <= 1100.0)
		tmp = ((2.0 + (s * ((s * 0.5) + 1.0))) ^ -c_p) / (0.5 ^ c_p);
	else
		tmp = (exp(t) + 1.0) ^ -c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 1100.0], N[(N[Power[N[(2.0 + N[(s * N[(N[(s * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Exp[t], $MachinePrecision] + 1.0), $MachinePrecision], (-c$95$p)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1100

   1. Initial program 94.1%

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

      1. associate-/l/ 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in c_n around 0 95.9%

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

      1. *-un-lft-identity 95.9%

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

      2. inv-pow 95.9%

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

      3. pow-pow 95.9%

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

      4. add-sqr-sqrt 48.8%

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

      5. sqrt-unprod 97.5%

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

      6. sqr-neg 97.5%

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

      7. sqrt-unprod 48.6%

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

      8. add-sqr-sqrt 97.1%

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

   7. Applied egg-rr 97.1%

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

      1. *-lft-identity 97.1%

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

      2. neg-mul-1 97.1%

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

   9. Simplified 97.1%

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

   10. Taylor expanded in t around 0 97.9%

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

   11. Taylor expanded in s around 0 97.9%

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

       1. *-commutative 97.9%

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

   13. Simplified 97.9%

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

   if 1100 < c_p

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in c_n around 0 25.2%

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

   6. Taylor expanded in c_p around 0 2.3%

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

      1. inv-pow 2.3%

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

      2. pow-pow 2.3%

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

   8. Applied egg-rr 75.8%

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

      1. *-commutative 75.8%

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

      2. neg-mul-1 75.8%

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

   10. Simplified 75.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 1100:\\ \;\;\;\;\frac{{\left(2 + s \cdot \left(s \cdot 0.5 + 1\right)\right)}^{\left(-c\_p\right)}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{t} + 1\right)}^{\left(-c\_p\right)}\\ \end{array} \]
5. 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 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 93.7%

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

   1. *-un-lft-identity 93.7%

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

   2. inv-pow 93.7%

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

   3. pow-pow 93.7%

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

   4. add-sqr-sqrt 47.3%

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

   5. sqrt-unprod 95.2%

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

   6. sqr-neg 95.2%

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

   7. sqrt-unprod 47.9%

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

   8. add-sqr-sqrt 94.8%

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

7. Applied egg-rr 94.8%

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

   1. *-lft-identity 94.8%

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

   2. neg-mul-1 94.8%

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

9. Simplified 94.8%

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

10. Taylor expanded in c_p around 0 94.1%

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

Developer Target 1: 96.2% 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 2024181 
(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))))