Harley's example

Percentage Accurate: 90.5% → 94.6%

Time: 58.4s

Alternatives: 7

Speedup: 835.0×

• could not determine a ground truth

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

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.5% 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: 94.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-t}\\ \mathbf{if}\;-t \leq 2 \cdot 10^{-237}:\\ \;\;\;\;\frac{{\left(e^{s} + 1\right)}^{\left(-c\_p\right)}}{{\left(\frac{1}{1 + t\_1}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - t\_1}\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- t))))
   (if (<= (- t) 2e-237)
     (/ (pow (+ (exp s) 1.0) (- c_p)) (pow (/ 1.0 (+ 1.0 t_1)) c_p))
     (/
      (pow (+ 1.0 (/ 1.0 (- -1.0 (exp (- s))))) c_n)
      (pow (+ 1.0 (/ 1.0 (- -1.0 t_1))) c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-t);
	double tmp;
	if (-t <= 2e-237) {
		tmp = pow((exp(s) + 1.0), -c_p) / pow((1.0 / (1.0 + t_1)), c_p);
	} else {
		tmp = pow((1.0 + (1.0 / (-1.0 - exp(-s)))), c_n) / pow((1.0 + (1.0 / (-1.0 - t_1))), 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) :: t_1
    real(8) :: tmp
    t_1 = exp(-t)
    if (-t <= 2d-237) then
        tmp = ((exp(s) + 1.0d0) ** -c_p) / ((1.0d0 / (1.0d0 + t_1)) ** c_p)
    else
        tmp = ((1.0d0 + (1.0d0 / ((-1.0d0) - exp(-s)))) ** c_n) / ((1.0d0 + (1.0d0 / ((-1.0d0) - t_1))) ** c_n)
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = Math.exp(-t);
	double tmp;
	if (-t <= 2e-237) {
		tmp = Math.pow((Math.exp(s) + 1.0), -c_p) / Math.pow((1.0 / (1.0 + t_1)), c_p);
	} else {
		tmp = Math.pow((1.0 + (1.0 / (-1.0 - Math.exp(-s)))), c_n) / Math.pow((1.0 + (1.0 / (-1.0 - t_1))), c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = math.exp(-t)
	tmp = 0
	if -t <= 2e-237:
		tmp = math.pow((math.exp(s) + 1.0), -c_p) / math.pow((1.0 / (1.0 + t_1)), c_p)
	else:
		tmp = math.pow((1.0 + (1.0 / (-1.0 - math.exp(-s)))), c_n) / math.pow((1.0 + (1.0 / (-1.0 - t_1))), c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-t))
	tmp = 0.0
	if (Float64(-t) <= 2e-237)
		tmp = Float64((Float64(exp(s) + 1.0) ^ Float64(-c_p)) / (Float64(1.0 / Float64(1.0 + t_1)) ^ c_p));
	else
		tmp = Float64((Float64(1.0 + Float64(1.0 / Float64(-1.0 - exp(Float64(-s))))) ^ c_n) / (Float64(1.0 + Float64(1.0 / Float64(-1.0 - t_1))) ^ c_n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	t_1 = exp(-t);
	tmp = 0.0;
	if (-t <= 2e-237)
		tmp = ((exp(s) + 1.0) ^ -c_p) / ((1.0 / (1.0 + t_1)) ^ c_p);
	else
		tmp = ((1.0 + (1.0 / (-1.0 - exp(-s)))) ^ c_n) / ((1.0 + (1.0 / (-1.0 - t_1))) ^ c_n);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-t)], $MachinePrecision]}, If[LessEqual[(-t), 2e-237], N[(N[Power[N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[N[(1.0 / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 + N[(1.0 / N[(-1.0 - N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[N[(1.0 + N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 2e-237

   1. Initial program 94.9%

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

      1. associate-/l/ 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in c_n around 0 97.3%

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

      1. *-un-lft-identity 97.3%

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

      2. inv-pow 97.3%

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

      3. pow-pow 97.3%

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

      4. add-sqr-sqrt 46.8%

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

      5. sqrt-unprod 98.6%

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

      6. sqr-neg 98.6%

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

      7. sqrt-unprod 51.8%

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

      8. add-sqr-sqrt 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. *-lft-identity 97.9%

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

      2. neg-mul-1 97.9%

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

   9. Simplified 97.9%

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

   if 2e-237 < (neg.f64 t)

   1. Initial program 91.1%

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

      1. associate-/l/ 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in c_p around 0 94.9%

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

   6. Taylor expanded in c_p around 0 99.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 2 \cdot 10^{-237}:\\ \;\;\;\;\frac{{\left(e^{s} + 1\right)}^{\left(-c\_p\right)}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ e^{\left(c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp
  (+
   (-
    (* c_n (- (log1p (/ 1.0 (+ (exp s) 1.0))) (log1p (/ 1.0 (+ 1.0 (exp t))))))
    (* c_p (log1p (exp s))))
   (* c_p (log1p (exp t))))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp((((c_n * (log1p((1.0 / (exp(s) + 1.0))) - log1p((1.0 / (1.0 + exp(t)))))) - (c_p * log1p(exp(s)))) + (c_p * log1p(exp(t)))));
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp((((c_n * (Math.log1p((1.0 / (Math.exp(s) + 1.0))) - Math.log1p((1.0 / (1.0 + Math.exp(t)))))) - (c_p * Math.log1p(Math.exp(s)))) + (c_p * Math.log1p(Math.exp(t)))));
}

Python

def code(c_p, c_n, t, s):
	return math.exp((((c_n * (math.log1p((1.0 / (math.exp(s) + 1.0))) - math.log1p((1.0 / (1.0 + math.exp(t)))))) - (c_p * math.log1p(math.exp(s)))) + (c_p * math.log1p(math.exp(t)))))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(Float64(c_n * Float64(log1p(Float64(1.0 / Float64(exp(s) + 1.0))) - log1p(Float64(1.0 / Float64(1.0 + exp(t)))))) - Float64(c_p * log1p(exp(s)))) + Float64(c_p * log1p(exp(t)))))
end

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. associate-/l/ 93.4%

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

3. Simplified 93.4%

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

6. Applied egg-rr 97.8%

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

   1. *-lft-identity 97.8%

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

   2. associate--l+ 97.9%

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

   3. distribute-lft-out-- 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 3: 93.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\frac{{\left(e^{s} + 1\right)}^{\left(-c\_p\right)}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot 0.5 + -1\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) -1e-307)
   (/ (pow (+ (exp s) 1.0) (- c_p)) (pow (/ 1.0 (+ 1.0 (exp (- t)))) c_p))
   (pow (/ 1.0 (+ 2.0 (* s (+ (* s 0.5) -1.0)))) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= -1e-307) {
		tmp = pow((exp(s) + 1.0), -c_p) / pow((1.0 / (1.0 + exp(-t))), c_p);
	} else {
		tmp = pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p);
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (-s <= (-1d-307)) then
        tmp = ((exp(s) + 1.0d0) ** -c_p) / ((1.0d0 / (1.0d0 + exp(-t))) ** c_p)
    else
        tmp = (1.0d0 / (2.0d0 + (s * ((s * 0.5d0) + (-1.0d0))))) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= -1e-307) {
		tmp = Math.pow((Math.exp(s) + 1.0), -c_p) / Math.pow((1.0 / (1.0 + Math.exp(-t))), c_p);
	} else {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= -1e-307:
		tmp = math.pow((math.exp(s) + 1.0), -c_p) / math.pow((1.0 / (1.0 + math.exp(-t))), c_p)
	else:
		tmp = math.pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= -1e-307)
		tmp = Float64((Float64(exp(s) + 1.0) ^ Float64(-c_p)) / (Float64(1.0 / Float64(1.0 + exp(Float64(-t)))) ^ c_p));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * 0.5) + -1.0)))) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= -1e-307)
		tmp = ((exp(s) + 1.0) ^ -c_p) / ((1.0 / (1.0 + exp(-t))) ^ c_p);
	else
		tmp = (1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), -1e-307], N[(N[Power[N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[N[(1.0 / N[(1.0 + N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(s * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < -9.99999999999999909e-308

   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. Step-by-step derivation

      1. associate-/l/ 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in c_n around 0 93.4%

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

      1. *-un-lft-identity 93.4%

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

      2. inv-pow 93.4%

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

      3. pow-pow 93.4%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 96.6%

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

      6. sqr-neg 96.6%

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

      7. sqrt-unprod 96.6%

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

      8. add-sqr-sqrt 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. *-lft-identity 96.6%

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

      2. neg-mul-1 96.6%

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

   9. Simplified 96.6%

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

   if -9.99999999999999909e-308 < (neg.f64 s)

   1. Initial program 94.8%

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

      1. associate-/l/ 94.8%

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

   3. Simplified 94.8%

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

   5. Taylor expanded in c_n around 0 95.6%

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

   6. Taylor expanded in c_p around 0 97.7%

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

   7. Taylor expanded in s around 0 99.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\frac{{\left(e^{s} + 1\right)}^{\left(-c\_p\right)}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot 0.5 + -1\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (pow
  (/ 1.0 (+ 2.0 (* s (+ (* s (+ 0.5 (* s -0.16666666666666666))) -1.0))))
  c_p))

C

double code(double c_p, double c_n, double t, double s) {
	return pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = (1.0d0 / (2.0d0 + (s * ((s * (0.5d0 + (s * (-0.16666666666666666d0)))) + (-1.0d0))))) ** c_p
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
}

Python

def code(c_p, c_n, t, s):
	return math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p)

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))) + -1.0)))) ^ c_p
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = (1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))) ^ c_p;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(s * N[(0.5 + N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]

Derivation

1. Initial program 93.4%

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

   1. associate-/l/ 93.4%

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

3. Simplified 93.4%

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

5. Taylor expanded in c_n around 0 94.6%

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

6. Taylor expanded in c_p around 0 93.9%

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

7. Taylor expanded in s around 0 96.1%

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

8. Final simplification 96.1%

   \[\leadsto {\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p} \]
9. Add Preprocessing

Alternative 5: 92.5% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = (1.0d0 / (2.0d0 + (s * ((s * 0.5d0) + (-1.0d0))))) ** c_p
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. associate-/l/ 93.4%

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

3. Simplified 93.4%

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

5. Taylor expanded in c_n around 0 94.6%

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

6. Taylor expanded in c_p around 0 93.9%

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

7. Taylor expanded in s around 0 95.7%

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

8. Final simplification 95.7%

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

Alternative 6: 93.5% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. associate-/l/ 93.4%

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

3. Simplified 93.4%

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

5. Taylor expanded in c_p around 0 93.9%

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

6. Taylor expanded in c_p around 0 96.1%

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

7. Taylor expanded in t around 0 95.3%

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

8. Taylor expanded in s around 0 95.4%

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

   1. *-commutative 95.4%

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

10. Simplified 95.4%

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

11. Final simplification 95.4%

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

Alternative 7: 93.5% 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 93.4%

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

   1. associate-/l/ 93.4%

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

3. Simplified 93.4%

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

5. Taylor expanded in c_n around 0 94.6%

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

6. Taylor expanded in c_p around 0 95.3%

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

Developer target: 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 2024108 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

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

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