Harley's example

Percentage Accurate: 90.9% → 96.7%

Time: 1.3min

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: 90.9% 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: 96.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+113} \lor \neg \left(-s \leq 2 \cdot 10^{-100}\right):\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(\sqrt[3]{t \cdot \mathsf{fma}\left(-0.5, c\_n, c\_p \cdot 0.5\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (or (<= (- s) -2e+113) (not (<= (- s) 2e-100)))
   (pow
    (/ 1.0 (+ 2.0 (* s (+ (* s (+ 0.5 (* s -0.16666666666666666))) -1.0))))
    c_p)
   (- 1.0 (pow (cbrt (* t (fma -0.5 c_n (* c_p 0.5)))) 3.0))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if ((-s <= -2e+113) || !(-s <= 2e-100)) {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	} else {
		tmp = 1.0 - pow(cbrt((t * fma(-0.5, c_n, (c_p * 0.5)))), 3.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if ((Float64(-s) <= -2e+113) || !(Float64(-s) <= 2e-100))
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	else
		tmp = Float64(1.0 - (cbrt(Float64(t * fma(-0.5, c_n, Float64(c_p * 0.5)))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[Or[LessEqual[(-s), -2e+113], N[Not[LessEqual[(-s), 2e-100]], $MachinePrecision]], 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], N[(1.0 - N[Power[N[Power[N[(t * N[(-0.5 * c$95$n + N[(c$95$p * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < -2e113 or 2e-100 < (neg.f64 s)

   1. Initial program 83.5%

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

      1. associate-/l/ 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in c_n around 0 79.7%

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

   6. Taylor expanded in s around 0 93.8%

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

   7. Taylor expanded in c_p around 0 100.0%

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

   if -2e113 < (neg.f64 s) < 2e-100

   1. Initial program 96.1%

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

      1. associate-/l/ 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in s around 0 97.1%

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

   6. Taylor expanded in t around 0 99.2%

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

      1. add-cube-cbrt 99.2%

         \[\leadsto 1 + -1 \cdot \color{blue}{\left(\left(\sqrt[3]{t \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)} \cdot \sqrt[3]{t \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)}\right) \cdot \sqrt[3]{t \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)}\right)} \]

      2. pow3 99.2%

         \[\leadsto 1 + -1 \cdot \color{blue}{{\left(\sqrt[3]{t \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)}\right)}^{3}} \]

      3. fma-define 99.2%

         \[\leadsto 1 + -1 \cdot {\left(\sqrt[3]{t \cdot \color{blue}{\mathsf{fma}\left(-0.5, c\_n, 0.5 \cdot c\_p\right)}}\right)}^{3} \]

      4. *-commutative 99.2%

         \[\leadsto 1 + -1 \cdot {\left(\sqrt[3]{t \cdot \mathsf{fma}\left(-0.5, c\_n, \color{blue}{c\_p \cdot 0.5}\right)}\right)}^{3} \]

   8. Applied egg-rr 99.2%

      \[\leadsto 1 + -1 \cdot \color{blue}{{\left(\sqrt[3]{t \cdot \mathsf{fma}\left(-0.5, c\_n, c\_p \cdot 0.5\right)}\right)}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+113} \lor \neg \left(-s \leq 2 \cdot 10^{-100}\right):\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;1 - {\left(\sqrt[3]{t \cdot \mathsf{fma}\left(-0.5, c\_n, c\_p \cdot 0.5\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-100}:\\ \;\;\;\;e^{\left(c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 2e-100)
   (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)))))
   (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) {
	double tmp;
	if (-s <= 2e-100) {
		tmp = 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)))));
	} else {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 2e-100) {
		tmp = 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)))));
	} else {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 2e-100:
		tmp = 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)))))
	else:
		tmp = math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 2e-100)
		tmp = 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)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 2e-100], 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], 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. Split input into 2 regimes

2. if (neg.f64 s) < 2e-100

   1. Initial program 94.4%

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

      1. associate-/l/ 94.4%

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

   3. Simplified 94.4%

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

   6. Applied egg-rr 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate--l+ 98.7%

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

      3. distribute-lft-out-- 98.7%

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

   8. Simplified 98.7%

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

   if 2e-100 < (neg.f64 s)

   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. Step-by-step derivation

      1. associate-/l/ 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in c_n around 0 92.8%

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

   6. Taylor expanded in s around 0 95.2%

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

   7. Taylor expanded in c_p around 0 100.0%

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + -0.16666666666666666 \cdot s\right) - 1\right)}\right)}^{c\_p}}{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-100}:\\ \;\;\;\;e^{\left(c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-t}\\ \mathbf{if}\;-t \leq 10^{-38}:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + t\_1}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 + \frac{1}{-1 - t\_1}\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- t))))
   (if (<= (- t) 1e-38)
     (/
      (pow
       (/ 1.0 (+ 2.0 (* s (+ (* s (+ 0.5 (* s -0.16666666666666666))) -1.0))))
       c_p)
      (pow (/ 1.0 (+ 1.0 t_1)) c_p))
     (/ (pow 0.5 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 <= 1e-38) {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p) / pow((1.0 / (1.0 + t_1)), c_p);
	} else {
		tmp = pow(0.5, 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 <= 1d-38) then
        tmp = ((1.0d0 / (2.0d0 + (s * ((s * (0.5d0 + (s * (-0.16666666666666666d0)))) + (-1.0d0))))) ** c_p) / ((1.0d0 / (1.0d0 + t_1)) ** c_p)
    else
        tmp = (0.5d0 ** 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 <= 1e-38) {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p) / Math.pow((1.0 / (1.0 + t_1)), c_p);
	} else {
		tmp = Math.pow(0.5, 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 <= 1e-38:
		tmp = math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p) / math.pow((1.0 / (1.0 + t_1)), c_p)
	else:
		tmp = math.pow(0.5, 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) <= 1e-38)
		tmp = Float64((Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))) + -1.0)))) ^ c_p) / (Float64(1.0 / Float64(1.0 + t_1)) ^ c_p));
	else
		tmp = Float64((0.5 ^ 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 <= 1e-38)
		tmp = ((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))) ^ c_p) / ((1.0 / (1.0 + t_1)) ^ c_p);
	else
		tmp = (0.5 ^ 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), 1e-38], N[(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] / N[Power[N[(1.0 / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision], N[(N[Power[0.5, 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) < 9.9999999999999996e-39

   1. Initial program 95.3%

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

      1. associate-/l/ 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in c_n around 0 95.5%

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

   6. Taylor expanded in s around 0 98.0%

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

   if 9.9999999999999996e-39 < (neg.f64 t)

   1. Initial program 76.6%

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

      1. associate-/l/ 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in s around 0 81.3%

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

   6. Taylor expanded in c_p around 0 100.0%

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

   7. Taylor expanded in c_p around 0 100.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 10^{-38}:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+113} \lor \neg \left(-s \leq 2 \cdot 10^{-100}\right):\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;1 - t \cdot \left(c\_n \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (or (<= (- s) -2e+113) (not (<= (- s) 2e-100)))
   (pow
    (/ 1.0 (+ 2.0 (* s (+ (* s (+ 0.5 (* s -0.16666666666666666))) -1.0))))
    c_p)
   (- 1.0 (* t (* c_n -0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if ((-s <= -2e+113) || !(-s <= 2e-100)) {
		tmp = pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	} else {
		tmp = 1.0 - (t * (c_n * -0.5));
	}
	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 <= (-2d+113)) .or. (.not. (-s <= 2d-100))) then
        tmp = (1.0d0 / (2.0d0 + (s * ((s * (0.5d0 + (s * (-0.16666666666666666d0)))) + (-1.0d0))))) ** c_p
    else
        tmp = 1.0d0 - (t * (c_n * (-0.5d0)))
    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 <= -2e+113) || !(-s <= 2e-100)) {
		tmp = Math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p);
	} else {
		tmp = 1.0 - (t * (c_n * -0.5));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if (-s <= -2e+113) or not (-s <= 2e-100):
		tmp = math.pow((1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))), c_p)
	else:
		tmp = 1.0 - (t * (c_n * -0.5))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if ((Float64(-s) <= -2e+113) || !(Float64(-s) <= 2e-100))
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(Float64(s * Float64(0.5 + Float64(s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	else
		tmp = Float64(1.0 - Float64(t * Float64(c_n * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if ((-s <= -2e+113) || ~((-s <= 2e-100)))
		tmp = (1.0 / (2.0 + (s * ((s * (0.5 + (s * -0.16666666666666666))) + -1.0)))) ^ c_p;
	else
		tmp = 1.0 - (t * (c_n * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[Or[LessEqual[(-s), -2e+113], N[Not[LessEqual[(-s), 2e-100]], $MachinePrecision]], 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], N[(1.0 - N[(t * N[(c$95$n * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < -2e113 or 2e-100 < (neg.f64 s)

   1. Initial program 83.5%

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

      1. associate-/l/ 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in c_n around 0 79.7%

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

   6. Taylor expanded in s around 0 93.8%

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

   7. Taylor expanded in c_p around 0 100.0%

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

   if -2e113 < (neg.f64 s) < 2e-100

   1. Initial program 96.1%

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

      1. associate-/l/ 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in s around 0 97.1%

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

   6. Taylor expanded in t around 0 99.2%

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

   7. Taylor expanded in c_n around inf 99.2%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

   9. Simplified 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq -2 \cdot 10^{+113} \lor \neg \left(-s \leq 2 \cdot 10^{-100}\right):\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(0.5 + s \cdot -0.16666666666666666\right) + -1\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;1 - t \cdot \left(c\_n \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 5 \cdot 10^{+113}:\\ \;\;\;\;1 - t \cdot \left(c\_n \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot 0.5\right)}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s 5e+113)
   (- 1.0 (* t (* c_n -0.5)))
   (pow (/ 1.0 (+ 2.0 (* s (+ -1.0 (* s 0.5))))) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= 5e+113) {
		tmp = 1.0 - (t * (c_n * -0.5));
	} else {
		tmp = pow((1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))), c_p);
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 5d+113) then
        tmp = 1.0d0 - (t * (c_n * (-0.5d0)))
    else
        tmp = (1.0d0 / (2.0d0 + (s * ((-1.0d0) + (s * 0.5d0))))) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= 5e+113) {
		tmp = 1.0 - (t * (c_n * -0.5));
	} else {
		tmp = Math.pow((1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= 5e+113:
		tmp = 1.0 - (t * (c_n * -0.5))
	else:
		tmp = math.pow((1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= 5e+113)
		tmp = Float64(1.0 - Float64(t * Float64(c_n * -0.5)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(s * Float64(-1.0 + Float64(s * 0.5))))) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= 5e+113)
		tmp = 1.0 - (t * (c_n * -0.5));
	else
		tmp = (1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, 5e+113], N[(1.0 - N[(t * N[(c$95$n * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 / N[(2.0 + N[(s * N[(-1.0 + N[(s * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 5e113

   1. Initial program 95.2%

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

      1. associate-/l/ 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in s around 0 94.4%

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

   6. Taylor expanded in t around 0 97.0%

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

   7. Taylor expanded in c_n around inf 97.0%

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

      1. associate-*r* 97.0%

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

      2. *-commutative 97.0%

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

   9. Simplified 97.0%

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

   if 5e113 < s

   1. Initial program 33.3%

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

      1. associate-/l/ 33.3%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in c_n around 0 2.9%

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

   6. Taylor expanded in s around 0 67.2%

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

   7. Taylor expanded in c_p around 0 83.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5 \cdot 10^{+113}:\\ \;\;\;\;1 - t \cdot \left(c\_n \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot 0.5\right)}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

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 - (t * (c_n * (-0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.8%

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

   1. associate-/l/ 93.8%

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

3. Simplified 93.8%

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

5. Taylor expanded in s around 0 92.3%

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

6. Taylor expanded in t around 0 94.8%

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

7. Taylor expanded in c_n around inf 94.8%

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

   1. associate-*r* 94.8%

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

   2. *-commutative 94.8%

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

9. Simplified 94.8%

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

10. Final simplification 94.8%

    \[\leadsto 1 - t \cdot \left(c\_n \cdot -0.5\right) \]
11. Add Preprocessing

Alternative 7: 94.2% 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 93.8%

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

   1. associate-/l/ 93.8%

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

3. Simplified 93.8%

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

5. Taylor expanded in c_n around 0 94.4%

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

6. Taylor expanded in c_p around 0 94.8%

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

7. Final simplification 94.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Developer target: 96.5% 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 2024073 
(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))))