Harley's example

Percentage Accurate: 91.3% → 96.8%

Time: 2.4min

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: 91.3% 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: 96.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 20:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{-1}{1 + e^{s}}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 20.0)
   (/
    (pow (/ 1.0 (+ 2.0 (* s (+ (* s (* s -0.16666666666666666)) -1.0)))) c_p)
    (pow 0.5 c_p))
   (/ (pow (+ 1.0 (/ -1.0 (+ 1.0 (exp s)))) c_n) (pow 0.5 c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 20.0) {
		tmp = pow((1.0 / (2.0 + (s * ((s * (s * -0.16666666666666666)) + -1.0)))), c_p) / pow(0.5, c_p);
	} else {
		tmp = pow((1.0 + (-1.0 / (1.0 + exp(s)))), c_n) / pow(0.5, c_n);
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (c_p <= 20.0d0) then
        tmp = ((1.0d0 / (2.0d0 + (s * ((s * (s * (-0.16666666666666666d0))) + (-1.0d0))))) ** c_p) / (0.5d0 ** c_p)
    else
        tmp = ((1.0d0 + ((-1.0d0) / (1.0d0 + exp(s)))) ** c_n) / (0.5d0 ** c_n)
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 20.0) {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * (s * -0.16666666666666666)) + -1.0)))), c_p) / Math.pow(0.5, c_p);
	} else {
		tmp = Math.pow((1.0 + (-1.0 / (1.0 + Math.exp(s)))), c_n) / Math.pow(0.5, c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 20.0:
		tmp = math.pow((1.0 / (2.0 + (s * ((s * (s * -0.16666666666666666)) + -1.0)))), c_p) / math.pow(0.5, c_p)
	else:
		tmp = math.pow((1.0 + (-1.0 / (1.0 + math.exp(s)))), c_n) / math.pow(0.5, c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 20.0)
		tmp = Float64((Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(s * -0.16666666666666666)) + -1.0)))) ^ c_p) / (0.5 ^ c_p));
	else
		tmp = Float64((Float64(1.0 + Float64(-1.0 / Float64(1.0 + exp(s)))) ^ c_n) / (0.5 ^ c_n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_p <= 20.0)
		tmp = ((1.0 / (2.0 + (s * ((s * (s * -0.16666666666666666)) + -1.0)))) ^ c_p) / (0.5 ^ c_p);
	else
		tmp = ((1.0 + (-1.0 / (1.0 + exp(s)))) ^ c_n) / (0.5 ^ c_n);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 20.0], N[(N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(s * N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] / N[Power[0.5, 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[0.5, c$95$n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 20

   1. Initial program 93.1%

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

      1. associate-/l/ 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in c_n around 0 94.0%

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

   6. Taylor expanded in t around 0 95.6%

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

   7. Taylor expanded in s around 0 97.6%

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

   8. Taylor expanded in s around inf 97.6%

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

      1. *-commutative 97.6%

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

   10. Simplified 97.6%

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

   if 20 < c_p

   1. Initial program 25.0%

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

      1. associate-/l/ 25.0%

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

   3. Simplified 25.0%

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

   5. Taylor expanded in c_p around 0 67.6%

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

   6. Taylor expanded in t around 0 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. sub-neg 67.7%

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

      3. distribute-neg-frac 67.7%

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

      4. metadata-eval 67.7%

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

      5. add-sqr-sqrt 25.8%

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

      6. sqrt-unprod 67.7%

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

      7. sqr-neg 67.7%

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

      8. sqrt-unprod 41.9%

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

      9. add-sqr-sqrt 91.9%

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

   8. Applied egg-rr 91.9%

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

      1. *-lft-identity 91.9%

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

   10. Simplified 91.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 20:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{-1}{1 + e^{s}}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 500:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 500.0)
   (/
    (pow (/ 1.0 (+ 2.0 (* s (+ (* s (* s -0.16666666666666666)) -1.0)))) c_p)
    (pow 0.5 c_p))
   1.0))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 500.0) {
		tmp = pow((1.0 / (2.0 + (s * ((s * (s * -0.16666666666666666)) + -1.0)))), c_p) / pow(0.5, c_p);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (c_p <= 500.0d0) then
        tmp = ((1.0d0 / (2.0d0 + (s * ((s * (s * (-0.16666666666666666d0))) + (-1.0d0))))) ** c_p) / (0.5d0 ** c_p)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 500.0) {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * (s * -0.16666666666666666)) + -1.0)))), c_p) / Math.pow(0.5, c_p);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 500.0:
		tmp = math.pow((1.0 / (2.0 + (s * ((s * (s * -0.16666666666666666)) + -1.0)))), c_p) / math.pow(0.5, c_p)
	else:
		tmp = 1.0
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 500.0)
		tmp = Float64((Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(s * -0.16666666666666666)) + -1.0)))) ^ c_p) / (0.5 ^ c_p));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_p <= 500.0)
		tmp = ((1.0 / (2.0 + (s * ((s * (s * -0.16666666666666666)) + -1.0)))) ^ c_p) / (0.5 ^ c_p);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 500.0], N[(N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(s * N[(s * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] / N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if c_p < 500

   1. Initial program 93.2%

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

      1. associate-/l/ 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in c_n around 0 94.1%

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

   6. Taylor expanded in t around 0 95.7%

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

   7. Taylor expanded in s around 0 97.6%

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

   8. Taylor expanded in s around inf 97.6%

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

      1. *-commutative 97.6%

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

   10. Simplified 97.6%

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

   if 500 < c_p

   1. Initial program 10.0%

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

      1. associate-/l/ 10.0%

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

   3. Simplified 10.0%

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

   5. Taylor expanded in c_p around 0 70.8%

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

   6. Taylor expanded in c_n around 0 70.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 500:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 880:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot 0.5 + -1\right)}\right)}^{c\_p}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 880.0)
   (/ (pow (/ 1.0 (+ 2.0 (* s (+ (* s 0.5) -1.0)))) c_p) (pow 0.5 c_p))
   1.0))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 880.0) {
		tmp = pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p) / pow(0.5, c_p);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (c_p <= 880.0d0) then
        tmp = ((1.0d0 / (2.0d0 + (s * ((s * 0.5d0) + (-1.0d0))))) ** c_p) / (0.5d0 ** c_p)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 880.0) {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p) / Math.pow(0.5, c_p);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 880.0:
		tmp = math.pow((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))), c_p) / math.pow(0.5, c_p)
	else:
		tmp = 1.0
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 880.0)
		tmp = Float64((Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * 0.5) + -1.0)))) ^ c_p) / (0.5 ^ c_p));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_p <= 880.0)
		tmp = ((1.0 / (2.0 + (s * ((s * 0.5) + -1.0)))) ^ c_p) / (0.5 ^ c_p);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 880.0], N[(N[Power[N[(1.0 / N[(2.0 + N[(s * N[(N[(s * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision] / N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if c_p < 880

   1. Initial program 93.2%

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

      1. associate-/l/ 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in c_n around 0 94.1%

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

   6. Taylor expanded in t around 0 95.7%

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

   7. Taylor expanded in s around 0 97.2%

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

   if 880 < c_p

   1. Initial program 10.0%

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

      1. associate-/l/ 10.0%

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

   3. Simplified 10.0%

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

   5. Taylor expanded in c_p around 0 70.8%

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

   6. Taylor expanded in c_n around 0 70.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 880:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot 0.5 + -1\right)}\right)}^{c\_p}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;1 + s \cdot \left(c\_n \cdot -0.5 + s \cdot \left(c\_n \cdot \left(c\_n \cdot 0.125 - 0.125\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= t -8.2e-10)
   (/ (pow 0.5 c_n) (pow (+ 0.5 (* t -0.25)) c_n))
   (+ 1.0 (* s (+ (* c_n -0.5) (* s (* c_n (- (* c_n 0.125) 0.125))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -8.2e-10) {
		tmp = pow(0.5, c_n) / pow((0.5 + (t * -0.25)), c_n);
	} else {
		tmp = 1.0 + (s * ((c_n * -0.5) + (s * (c_n * ((c_n * 0.125) - 0.125)))));
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (t <= (-8.2d-10)) then
        tmp = (0.5d0 ** c_n) / ((0.5d0 + (t * (-0.25d0))) ** c_n)
    else
        tmp = 1.0d0 + (s * ((c_n * (-0.5d0)) + (s * (c_n * ((c_n * 0.125d0) - 0.125d0)))))
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -8.2e-10) {
		tmp = Math.pow(0.5, c_n) / Math.pow((0.5 + (t * -0.25)), c_n);
	} else {
		tmp = 1.0 + (s * ((c_n * -0.5) + (s * (c_n * ((c_n * 0.125) - 0.125)))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if t <= -8.2e-10:
		tmp = math.pow(0.5, c_n) / math.pow((0.5 + (t * -0.25)), c_n)
	else:
		tmp = 1.0 + (s * ((c_n * -0.5) + (s * (c_n * ((c_n * 0.125) - 0.125)))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (t <= -8.2e-10)
		tmp = Float64((0.5 ^ c_n) / (Float64(0.5 + Float64(t * -0.25)) ^ c_n));
	else
		tmp = Float64(1.0 + Float64(s * Float64(Float64(c_n * -0.5) + Float64(s * Float64(c_n * Float64(Float64(c_n * 0.125) - 0.125))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (t <= -8.2e-10)
		tmp = (0.5 ^ c_n) / ((0.5 + (t * -0.25)) ^ c_n);
	else
		tmp = 1.0 + (s * ((c_n * -0.5) + (s * (c_n * ((c_n * 0.125) - 0.125)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[t, -8.2e-10], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(0.5 + N[(t * -0.25), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(s * N[(N[(c$95$n * -0.5), $MachinePrecision] + N[(s * N[(c$95$n * N[(N[(c$95$n * 0.125), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.1999999999999996e-10

   1. Initial program 31.5%

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

      1. associate-/l/ 31.5%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in c_p around 0 85.1%

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

   6. Taylor expanded in s around 0 92.5%

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

   7. Taylor expanded in t around 0 92.5%

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

      1. *-commutative 92.5%

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

   9. Simplified 92.5%

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

   if -8.1999999999999996e-10 < t

   1. Initial program 93.1%

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

      1. associate-/l/ 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in c_p around 0 94.7%

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

   6. Taylor expanded in t around 0 95.1%

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

   7. Taylor expanded in s around 0 96.0%

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

   8. Taylor expanded in c_n around 0 96.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;1 + s \cdot \left(c\_n \cdot -0.5 + s \cdot \left(c\_n \cdot \left(c\_n \cdot 0.125 - 0.125\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.0% accurate, 49.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = 1.0d0 + (s * ((c_n * (-0.5d0)) + (s * (c_n * ((c_n * 0.125d0) - 0.125d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. associate-/l/ 89.9%

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

3. Simplified 89.9%

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

5. Taylor expanded in c_p around 0 94.2%

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

6. Taylor expanded in t around 0 92.6%

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

7. Taylor expanded in s around 0 93.6%

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

8. Taylor expanded in c_n around 0 93.6%

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

9. Final simplification 93.6%

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

Alternative 6: 94.1% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. associate-/l/ 89.9%

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

3. Simplified 89.9%

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

5. Taylor expanded in c_p around 0 94.2%

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

6. Taylor expanded in t around 0 92.6%

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

7. Taylor expanded in s around 0 93.5%

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

8. Final simplification 93.5%

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

Alternative 7: 94.1% accurate, 835.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. associate-/l/ 89.9%

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

3. Simplified 89.9%

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

5. Taylor expanded in c_p around 0 94.2%

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

6. Taylor expanded in c_n around 0 93.5%

   \[\leadsto \color{blue}{1} \]
7. 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 2024156 
(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))))