Harley's example

Percentage Accurate: 90.9% → 97.7%

Time: 52.0s

Alternatives: 8

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 2.360758581390745e-40
  2. c_n = 1.3401192833948233e-301
  3. t = -1.1977662533671944e-29
  4. s = 1.061871156409709e+167

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: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + e^{0 - s}\\ \mathbf{if}\;0 - s \leq -5 \cdot 10^{+28}:\\ \;\;\;\;{\left(1 + \frac{-1}{t\_1}\right)}^{c\_n}\\ \mathbf{elif}\;0 - s \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{t\_1}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (+ 1.0 (exp (- 0.0 s)))))
   (if (<= (- 0.0 s) -5e+28)
     (pow (+ 1.0 (/ -1.0 t_1)) c_n)
     (if (<= (- 0.0 s) 50000000.0) 1.0 (pow (/ 1.0 t_1) c_p)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 + exp((0.0 - s));
	double tmp;
	if ((0.0 - s) <= -5e+28) {
		tmp = pow((1.0 + (-1.0 / t_1)), c_n);
	} else if ((0.0 - s) <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = pow((1.0 / t_1), 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) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + exp((0.0d0 - s))
    if ((0.0d0 - s) <= (-5d+28)) then
        tmp = (1.0d0 + ((-1.0d0) / t_1)) ** c_n
    else if ((0.0d0 - s) <= 50000000.0d0) then
        tmp = 1.0d0
    else
        tmp = (1.0d0 / t_1) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 + Math.exp((0.0 - s));
	double tmp;
	if ((0.0 - s) <= -5e+28) {
		tmp = Math.pow((1.0 + (-1.0 / t_1)), c_n);
	} else if ((0.0 - s) <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = Math.pow((1.0 / t_1), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = 1.0 + math.exp((0.0 - s))
	tmp = 0
	if (0.0 - s) <= -5e+28:
		tmp = math.pow((1.0 + (-1.0 / t_1)), c_n)
	elif (0.0 - s) <= 50000000.0:
		tmp = 1.0
	else:
		tmp = math.pow((1.0 / t_1), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 + exp(Float64(0.0 - s)))
	tmp = 0.0
	if (Float64(0.0 - s) <= -5e+28)
		tmp = Float64(1.0 + Float64(-1.0 / t_1)) ^ c_n;
	elseif (Float64(0.0 - s) <= 50000000.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / t_1) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	t_1 = 1.0 + exp((0.0 - s));
	tmp = 0.0;
	if ((0.0 - s) <= -5e+28)
		tmp = (1.0 + (-1.0 / t_1)) ^ c_n;
	elseif ((0.0 - s) <= 50000000.0)
		tmp = 1.0;
	else
		tmp = (1.0 / t_1) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 + N[Exp[N[(0.0 - s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(0.0 - s), $MachinePrecision], -5e+28], N[Power[N[(1.0 + N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision], If[LessEqual[N[(0.0 - s), $MachinePrecision], 50000000.0], 1.0, N[Power[N[(1.0 / t$95$1), $MachinePrecision], c$95$p], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (neg.f64 s) < -4.99999999999999957e28

   1. Initial program 12.5%

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

   3. Taylor expanded in c_n around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right), c\_p\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 62.5%

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

   6. Taylor expanded in c_p around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_n}\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}} \]

   if -4.99999999999999957e28 < (neg.f64 s) < 5e7

   1. Initial program 91.8%

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

   3. Taylor expanded in c_p around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}\right), \color{blue}{\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_n\right), \left({\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_n}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      9. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_n}\right)\right) \]

   5. Simplified 96.7%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + \frac{1}{e^{s}}}\right)}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

   6. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 98.4%

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

   if 5e7 < (neg.f64 s)

   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. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right)\right), c\_n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in c_n around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_p}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \frac{1}{e^{s}}\right)\right), c\_p\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right) \]

      7. rec-exp N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{-1 \cdot s}\right)\right)\right), c\_p\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-1 \cdot s\right)\right)\right)\right), c\_p\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), c\_p\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - s\right)\right)\right)\right), c\_p\right) \]

      12. --lowering--.f64 100.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, s\right)\right)\right)\right), c\_p\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{0 - s}}\right)}^{c\_p}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - s \leq -5 \cdot 10^{+28}:\\ \;\;\;\;{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}\\ \mathbf{elif}\;0 - s \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{1 + e^{0 - s}}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -750000000:\\ \;\;\;\;{\left(\frac{1}{1 + e^{0 - s}}\right)}^{c\_p}\\ \mathbf{elif}\;s \leq 2.15 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_n}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -750000000.0)
   (pow (/ 1.0 (+ 1.0 (exp (- 0.0 s)))) c_p)
   (if (<= s 2.15e-13) 1.0 (pow 0.5 c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -750000000.0) {
		tmp = pow((1.0 / (1.0 + exp((0.0 - s)))), c_p);
	} else if (s <= 2.15e-13) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-750000000.0d0)) then
        tmp = (1.0d0 / (1.0d0 + exp((0.0d0 - s)))) ** c_p
    else if (s <= 2.15d-13) then
        tmp = 1.0d0
    else
        tmp = 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 <= -750000000.0) {
		tmp = Math.pow((1.0 / (1.0 + Math.exp((0.0 - s)))), c_p);
	} else if (s <= 2.15e-13) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(0.5, c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -750000000.0:
		tmp = math.pow((1.0 / (1.0 + math.exp((0.0 - s)))), c_p)
	elif s <= 2.15e-13:
		tmp = 1.0
	else:
		tmp = math.pow(0.5, c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -750000000.0)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(0.0 - s)))) ^ c_p;
	elseif (s <= 2.15e-13)
		tmp = 1.0;
	else
		tmp = 0.5 ^ c_n;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -750000000.0)
		tmp = (1.0 / (1.0 + exp((0.0 - s)))) ^ c_p;
	elseif (s <= 2.15e-13)
		tmp = 1.0;
	else
		tmp = 0.5 ^ c_n;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -750000000.0], N[Power[N[(1.0 / N[(1.0 + N[Exp[N[(0.0 - s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], If[LessEqual[s, 2.15e-13], 1.0, N[Power[0.5, c$95$n], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if s < -7.5e8

   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. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right)\right), c\_n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in c_n around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_p}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \frac{1}{e^{s}}\right)\right), c\_p\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right) \]

      7. rec-exp N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{-1 \cdot s}\right)\right)\right), c\_p\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-1 \cdot s\right)\right)\right)\right), c\_p\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), c\_p\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - s\right)\right)\right)\right), c\_p\right) \]

      12. --lowering--.f64 100.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, s\right)\right)\right)\right), c\_p\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{0 - s}}\right)}^{c\_p}} \]

   if -7.5e8 < s < 2.1499999999999999e-13

   1. Initial program 91.9%

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

   3. Taylor expanded in c_p around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}\right), \color{blue}{\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_n\right), \left({\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_n}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      9. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_n}\right)\right) \]

   5. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + \frac{1}{e^{s}}}\right)}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

   6. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 98.3%

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

   if 2.1499999999999999e-13 < s

   1. Initial program 55.7%

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

   3. Taylor expanded in c_n around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right), c\_p\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.0%

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

   6. Taylor expanded in c_p around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_n}\right) \]

   8. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \color{blue}{{\frac{1}{2}}^{c\_n}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 89.4%

          \[\leadsto \mathsf{pow.f64}\left(\frac{1}{2}, \color{blue}{c\_n}\right) \]

   11. Simplified 89.4%

       \[\leadsto \color{blue}{{0.5}^{c\_n}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.1% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -2000000000:\\ \;\;\;\;{\left(\frac{\frac{-6}{s \cdot s}}{s}\right)}^{c\_p}\\ \mathbf{elif}\;s \leq 5 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_n}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -2000000000.0)
   (pow (/ (/ -6.0 (* s s)) s) c_p)
   (if (<= s 5e-13) 1.0 (pow 0.5 c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -2000000000.0) {
		tmp = pow(((-6.0 / (s * s)) / s), c_p);
	} else if (s <= 5e-13) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-2000000000.0d0)) then
        tmp = (((-6.0d0) / (s * s)) / s) ** c_p
    else if (s <= 5d-13) then
        tmp = 1.0d0
    else
        tmp = 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 <= -2000000000.0) {
		tmp = Math.pow(((-6.0 / (s * s)) / s), c_p);
	} else if (s <= 5e-13) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(0.5, c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -2000000000.0:
		tmp = math.pow(((-6.0 / (s * s)) / s), c_p)
	elif s <= 5e-13:
		tmp = 1.0
	else:
		tmp = math.pow(0.5, c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -2000000000.0)
		tmp = Float64(Float64(-6.0 / Float64(s * s)) / s) ^ c_p;
	elseif (s <= 5e-13)
		tmp = 1.0;
	else
		tmp = 0.5 ^ c_n;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -2000000000.0)
		tmp = ((-6.0 / (s * s)) / s) ^ c_p;
	elseif (s <= 5e-13)
		tmp = 1.0;
	else
		tmp = 0.5 ^ c_n;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -2000000000.0], N[Power[N[(N[(-6.0 / N[(s * s), $MachinePrecision]), $MachinePrecision] / s), $MachinePrecision], c$95$p], $MachinePrecision], If[LessEqual[s, 5e-13], 1.0, N[Power[0.5, c$95$n], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if s < -2e9

   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. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right)\right), c\_n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in c_n around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_p}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \frac{1}{e^{s}}\right)\right), c\_p\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right) \]

      7. rec-exp N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{-1 \cdot s}\right)\right)\right), c\_p\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-1 \cdot s\right)\right)\right)\right), c\_p\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), c\_p\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - s\right)\right)\right)\right), c\_p\right) \]

      12. --lowering--.f64 100.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, s\right)\right)\right)\right), c\_p\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{0 - s}}\right)}^{c\_p}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(2 + s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right)\right)}\right), c\_p\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right)\right)\right)\right), c\_p\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right)\right)\right)\right), c\_p\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), c\_p\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) + -1\right)\right)\right)\right), c\_p\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \left(-1 + s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right)\right)\right)\right)\right), c\_p\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \mathsf{+.f64}\left(-1, \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right)\right)\right)\right)\right)\right), c\_p\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(s, \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right)\right)\right)\right)\right)\right), c\_p\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(s, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot s\right)\right)\right)\right)\right)\right)\right), c\_p\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(s, \mathsf{+.f64}\left(\frac{1}{2}, \left(s \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), c\_p\right) \]

       10. *-lowering-*.f64 86.2%

           \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(s, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(s, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right), c\_p\right) \]

   11. Simplified 86.2%

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

   12. Taylor expanded in s around inf

       \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left(\frac{-6}{{s}^{3}}\right)}, c\_p\right) \]
   13. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(\frac{-6}{\left(s \cdot s\right) \cdot s}\right), c\_p\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(\frac{-6}{{s}^{2} \cdot s}\right), c\_p\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\frac{-6}{{s}^{2}}}{s}\right), c\_p\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\frac{-6}{{s}^{2}}\right), s\right), c\_p\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-6, \left({s}^{2}\right)\right), s\right), c\_p\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-6, \left(s \cdot s\right)\right), s\right), c\_p\right) \]

       7. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-6, \mathsf{*.f64}\left(s, s\right)\right), s\right), c\_p\right) \]

   14. Simplified 86.2%

       \[\leadsto {\color{blue}{\left(\frac{\frac{-6}{s \cdot s}}{s}\right)}}^{c\_p} \]

   if -2e9 < s < 4.9999999999999999e-13

   1. Initial program 91.9%

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

   3. Taylor expanded in c_p around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}\right), \color{blue}{\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_n\right), \left({\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_n}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      9. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_n}\right)\right) \]

   5. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + \frac{1}{e^{s}}}\right)}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

   6. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 98.3%

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

   if 4.9999999999999999e-13 < s

   1. Initial program 55.7%

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

   3. Taylor expanded in c_n around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right), c\_p\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.0%

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

   6. Taylor expanded in c_p around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_n}\right) \]

   8. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \color{blue}{{\frac{1}{2}}^{c\_n}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 89.4%

          \[\leadsto \mathsf{pow.f64}\left(\frac{1}{2}, \color{blue}{c\_n}\right) \]

   11. Simplified 89.4%

       \[\leadsto \color{blue}{{0.5}^{c\_n}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -29000000:\\ \;\;\;\;{\left(\frac{2}{s \cdot s}\right)}^{c\_p}\\ \mathbf{elif}\;s \leq 5 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_n}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -29000000.0)
   (pow (/ 2.0 (* s s)) c_p)
   (if (<= s 5e-14) 1.0 (pow 0.5 c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -29000000.0) {
		tmp = pow((2.0 / (s * s)), c_p);
	} else if (s <= 5e-14) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-29000000.0d0)) then
        tmp = (2.0d0 / (s * s)) ** c_p
    else if (s <= 5d-14) then
        tmp = 1.0d0
    else
        tmp = 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 <= -29000000.0) {
		tmp = Math.pow((2.0 / (s * s)), c_p);
	} else if (s <= 5e-14) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(0.5, c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -29000000.0:
		tmp = math.pow((2.0 / (s * s)), c_p)
	elif s <= 5e-14:
		tmp = 1.0
	else:
		tmp = math.pow(0.5, c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -29000000.0)
		tmp = Float64(2.0 / Float64(s * s)) ^ c_p;
	elseif (s <= 5e-14)
		tmp = 1.0;
	else
		tmp = 0.5 ^ c_n;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -29000000.0)
		tmp = (2.0 / (s * s)) ^ c_p;
	elseif (s <= 5e-14)
		tmp = 1.0;
	else
		tmp = 0.5 ^ c_n;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -29000000.0], N[Power[N[(2.0 / N[(s * s), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], If[LessEqual[s, 5e-14], 1.0, N[Power[0.5, c$95$n], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if s < -2.9e7

   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. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right)\right), c\_n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in c_n around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_p}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \frac{1}{e^{s}}\right)\right), c\_p\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right) \]

      7. rec-exp N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{-1 \cdot s}\right)\right)\right), c\_p\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-1 \cdot s\right)\right)\right)\right), c\_p\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), c\_p\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - s\right)\right)\right)\right), c\_p\right) \]

      12. --lowering--.f64 100.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, s\right)\right)\right)\right), c\_p\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{0 - s}}\right)}^{c\_p}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)\right)}\right), c\_p\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(s \cdot \left(\frac{1}{2} \cdot s - 1\right)\right)\right)\right), c\_p\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \left(\frac{1}{2} \cdot s - 1\right)\right)\right)\right), c\_p\right) \]

       3. sub-neg N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \left(\frac{1}{2} \cdot s + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), c\_p\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \left(\frac{1}{2} \cdot s + -1\right)\right)\right)\right), c\_p\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \left(-1 + \frac{1}{2} \cdot s\right)\right)\right)\right), c\_p\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \mathsf{+.f64}\left(-1, \left(\frac{1}{2} \cdot s\right)\right)\right)\right)\right), c\_p\right) \]

       7. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(s, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{1}{2}, s\right)\right)\right)\right)\right), c\_p\right) \]

   11. Simplified 86.2%

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

   12. Taylor expanded in s around inf

       \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left(\frac{2}{{s}^{2}}\right)}, c\_p\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(2, \left({s}^{2}\right)\right), c\_p\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(2, \left(s \cdot s\right)\right), c\_p\right) \]

       3. *-lowering-*.f64 86.2%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(s, s\right)\right), c\_p\right) \]

   14. Simplified 86.2%

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

   if -2.9e7 < s < 5.0000000000000002e-14

   1. Initial program 91.9%

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

   3. Taylor expanded in c_p around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}\right), \color{blue}{\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_n\right), \left({\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_n}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      9. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_n}\right)\right) \]

   5. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + \frac{1}{e^{s}}}\right)}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

   6. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 98.3%

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

   if 5.0000000000000002e-14 < s

   1. Initial program 55.7%

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

   3. Taylor expanded in c_n around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right), c\_p\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.0%

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

   6. Taylor expanded in c_p around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_n}\right) \]

   8. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \color{blue}{{\frac{1}{2}}^{c\_n}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 89.4%

          \[\leadsto \mathsf{pow.f64}\left(\frac{1}{2}, \color{blue}{c\_n}\right) \]

   11. Simplified 89.4%

       \[\leadsto \color{blue}{{0.5}^{c\_n}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -40000000:\\ \;\;\;\;{\left(\frac{-1}{s}\right)}^{c\_p}\\ \mathbf{elif}\;s \leq 4 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_n}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -40000000.0)
   (pow (/ -1.0 s) c_p)
   (if (<= s 4e-13) 1.0 (pow 0.5 c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -40000000.0) {
		tmp = pow((-1.0 / s), c_p);
	} else if (s <= 4e-13) {
		tmp = 1.0;
	} else {
		tmp = 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 <= (-40000000.0d0)) then
        tmp = ((-1.0d0) / s) ** c_p
    else if (s <= 4d-13) then
        tmp = 1.0d0
    else
        tmp = 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 <= -40000000.0) {
		tmp = Math.pow((-1.0 / s), c_p);
	} else if (s <= 4e-13) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(0.5, c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -40000000.0:
		tmp = math.pow((-1.0 / s), c_p)
	elif s <= 4e-13:
		tmp = 1.0
	else:
		tmp = math.pow(0.5, c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -40000000.0)
		tmp = Float64(-1.0 / s) ^ c_p;
	elseif (s <= 4e-13)
		tmp = 1.0;
	else
		tmp = 0.5 ^ c_n;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -40000000.0)
		tmp = (-1.0 / s) ^ c_p;
	elseif (s <= 4e-13)
		tmp = 1.0;
	else
		tmp = 0.5 ^ c_n;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -40000000.0], N[Power[N[(-1.0 / s), $MachinePrecision], c$95$p], $MachinePrecision], If[LessEqual[s, 4e-13], 1.0, N[Power[0.5, c$95$n], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if s < -4e7

   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. Add Preprocessing

   3. Taylor expanded in c_p around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right)\right), c\_n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in c_n around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_p}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \frac{1}{e^{s}}\right)\right), c\_p\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right) \]

      7. rec-exp N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{-1 \cdot s}\right)\right)\right), c\_p\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-1 \cdot s\right)\right)\right)\right), c\_p\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), c\_p\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - s\right)\right)\right)\right), c\_p\right) \]

      12. --lowering--.f64 100.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, s\right)\right)\right)\right), c\_p\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{0 - s}}\right)}^{c\_p}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(2 + -1 \cdot s\right)}\right), c\_p\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(\mathsf{neg}\left(s\right)\right)\right)\right), c\_p\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(2 - s\right)\right), c\_p\right) \]

       3. --lowering--.f64 72.3%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(2, s\right)\right), c\_p\right) \]

   11. Simplified 72.3%

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

   12. Taylor expanded in s around -inf

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

       1. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(\frac{-1}{s}\right), \color{blue}{c\_p}\right) \]

       2. /-lowering-/.f64 72.3%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(-1, s\right), c\_p\right) \]

   14. Simplified 72.3%

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

   if -4e7 < s < 4.0000000000000001e-13

   1. Initial program 91.9%

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

   3. Taylor expanded in c_p around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}\right), \color{blue}{\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_n\right), \left({\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_n}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      9. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_n}\right)\right) \]

   5. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + \frac{1}{e^{s}}}\right)}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

   6. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 98.3%

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

   if 4.0000000000000001e-13 < s

   1. Initial program 55.7%

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

   3. Taylor expanded in c_n around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right), c\_p\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.0%

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

   6. Taylor expanded in c_p around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_n}\right) \]

   8. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \color{blue}{{\frac{1}{2}}^{c\_n}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 89.4%

          \[\leadsto \mathsf{pow.f64}\left(\frac{1}{2}, \color{blue}{c\_n}\right) \]

   11. Simplified 89.4%

       \[\leadsto \color{blue}{{0.5}^{c\_n}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 92.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-101}:\\ \;\;\;\;{0.5}^{c\_n}\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= t 5e-101) (pow 0.5 c_n) (pow 0.5 c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= 5e-101) {
		tmp = pow(0.5, c_n);
	} else {
		tmp = 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 (t <= 5d-101) then
        tmp = 0.5d0 ** c_n
    else
        tmp = 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 (t <= 5e-101) {
		tmp = Math.pow(0.5, c_n);
	} else {
		tmp = Math.pow(0.5, c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if t <= 5e-101:
		tmp = math.pow(0.5, c_n)
	else:
		tmp = math.pow(0.5, c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (t <= 5e-101)
		tmp = 0.5 ^ c_n;
	else
		tmp = 0.5 ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (t <= 5e-101)
		tmp = 0.5 ^ c_n;
	else
		tmp = 0.5 ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[t, 5e-101], N[Power[0.5, c$95$n], $MachinePrecision], N[Power[0.5, c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.0000000000000001e-101

   1. Initial program 89.6%

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

   3. Taylor expanded in c_n around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right), c\_p\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 89.3%

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

   6. Taylor expanded in c_p around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_n}\right) \]

   8. Simplified 95.0%

      \[\leadsto \color{blue}{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \color{blue}{{\frac{1}{2}}^{c\_n}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 94.5%

          \[\leadsto \mathsf{pow.f64}\left(\frac{1}{2}, \color{blue}{c\_n}\right) \]

   11. Simplified 94.5%

       \[\leadsto \color{blue}{{0.5}^{c\_n}} \]

   if 5.0000000000000001e-101 < t

   1. Initial program 80.2%

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

   3. Taylor expanded in c_p around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right)\right), c\_n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 82.4%

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

   6. Taylor expanded in c_n around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_p}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \frac{1}{e^{s}}\right)\right), c\_p\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right) \]

      7. rec-exp N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{-1 \cdot s}\right)\right)\right), c\_p\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-1 \cdot s\right)\right)\right)\right), c\_p\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), c\_p\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - s\right)\right)\right)\right), c\_p\right) \]

      12. --lowering--.f64 95.7%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, s\right)\right)\right)\right), c\_p\right) \]

   8. Simplified 95.7%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{0 - s}}\right)}^{c\_p}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \color{blue}{{\frac{1}{2}}^{c\_p}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 97.8%

          \[\leadsto \mathsf{pow.f64}\left(\frac{1}{2}, \color{blue}{c\_p}\right) \]

   11. Simplified 97.8%

       \[\leadsto \color{blue}{{0.5}^{c\_p}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 95.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 7.5 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_n}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s 7.5e-13) 1.0 (pow 0.5 c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= 7.5e-13) {
		tmp = 1.0;
	} else {
		tmp = 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 <= 7.5d-13) then
        tmp = 1.0d0
    else
        tmp = 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 <= 7.5e-13) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(0.5, c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= 7.5e-13:
		tmp = 1.0
	else:
		tmp = math.pow(0.5, c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= 7.5e-13)
		tmp = 1.0;
	else
		tmp = 0.5 ^ c_n;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= 7.5e-13)
		tmp = 1.0;
	else
		tmp = 0.5 ^ c_n;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, 7.5e-13], 1.0, N[Power[0.5, c$95$n], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 7.5000000000000004e-13

   1. Initial program 90.4%

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

   3. Taylor expanded in c_p around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}\right), \color{blue}{\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_n\right), \left({\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_n}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      9. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_n}\right)\right) \]

   5. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + \frac{1}{e^{s}}}\right)}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

   6. Taylor expanded in c_n around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 95.5%

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

   if 7.5000000000000004e-13 < s

   1. Initial program 55.7%

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

   3. Taylor expanded in c_n around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right), c\_p\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 78.0%

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

   6. Taylor expanded in c_p around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_n}\right) \]

   8. Simplified 94.6%

      \[\leadsto \color{blue}{{\left(1 + \frac{-1}{1 + e^{0 - s}}\right)}^{c\_n}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \color{blue}{{\frac{1}{2}}^{c\_n}} \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 89.4%

          \[\leadsto \mathsf{pow.f64}\left(\frac{1}{2}, \color{blue}{c\_n}\right) \]

   11. Simplified 89.4%

       \[\leadsto \color{blue}{{0.5}^{c\_n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.3% accurate, 835.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in c_p around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}\right), \color{blue}{\left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)}\right) \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_n\right), \left({\color{blue}{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_n}\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right), c\_n\right), \left({\left(\color{blue}{1} - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   9. exp-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \left({\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}\right)\right) \]

   12. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(-1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right)\right), c\_n\right), \mathsf{pow.f64}\left(\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_n}\right)\right) \]

5. Simplified 92.7%

   \[\leadsto \color{blue}{\frac{{\left(1 + \frac{-1}{1 + \frac{1}{e^{s}}}\right)}^{c\_n}}{{\left(1 + \frac{-1}{1 + e^{0 - t}}\right)}^{c\_n}}} \]

6. Taylor expanded in c_n around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 92.8%

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

Developer Target 1: 96.4% 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 2024192 
(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))))