Harley's example

Percentage Accurate: 90.6% → 98.4%

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: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + e^{t}\\ \mathbf{if}\;c\_p \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right), c\_n, \mathsf{log1p}\left(e^{s}\right) \cdot \left(-c\_p\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{t\_1}\right)\right)}}{{t\_1}^{\left(-c\_p\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left({e}^{c\_p}\right)}^{\left(\mathsf{log1p}\left(e^{-t}\right) + \left(s \cdot \left(0.5 + s \cdot -0.125\right) - \log 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (+ 1.0 (exp t))))
   (if (<= c_p 5e-85)
     (/
      (exp
       (fma
        (log1p (/ 1.0 (+ 1.0 (exp s))))
        c_n
        (- (* (log1p (exp s)) (- c_p)) (* c_n (log1p (/ 1.0 t_1))))))
      (pow t_1 (- c_p)))
     (pow
      (pow E c_p)
      (+ (log1p (exp (- t))) (- (* s (+ 0.5 (* s -0.125))) (log 2.0)))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = 1.0 + exp(t);
	double tmp;
	if (c_p <= 5e-85) {
		tmp = exp(fma(log1p((1.0 / (1.0 + exp(s)))), c_n, ((log1p(exp(s)) * -c_p) - (c_n * log1p((1.0 / t_1)))))) / pow(t_1, -c_p);
	} else {
		tmp = pow(pow(((double) M_E), c_p), (log1p(exp(-t)) + ((s * (0.5 + (s * -0.125))) - log(2.0))));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(1.0 + exp(t))
	tmp = 0.0
	if (c_p <= 5e-85)
		tmp = Float64(exp(fma(log1p(Float64(1.0 / Float64(1.0 + exp(s)))), c_n, Float64(Float64(log1p(exp(s)) * Float64(-c_p)) - Float64(c_n * log1p(Float64(1.0 / t_1)))))) / (t_1 ^ Float64(-c_p)));
	else
		tmp = (exp(1) ^ c_p) ^ Float64(log1p(exp(Float64(-t))) + Float64(Float64(s * Float64(0.5 + Float64(s * -0.125))) - log(2.0)));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(1.0 + N[Exp[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$p, 5e-85], N[(N[Exp[N[(N[Log[1 + N[(1.0 / N[(1.0 + N[Exp[s], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c$95$n + N[(N[(N[Log[1 + N[Exp[s], $MachinePrecision]], $MachinePrecision] * (-c$95$p)), $MachinePrecision] - N[(c$95$n * N[Log[1 + N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[t$95$1, (-c$95$p)], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[E, c$95$p], $MachinePrecision], N[(N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision] + N[(N[(s * N[(0.5 + N[(s * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 5.0000000000000002e-85

   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

   6. Applied egg-rr 98.3%

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

      1. associate-*l/ 98.3%

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

   8. Applied egg-rr 99.9%

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

      1. fma-define 99.9%

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

   10. Simplified 99.9%

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

   if 5.0000000000000002e-85 < c_p

   1. Initial program 85.1%

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

      1. associate-/l/ 85.1%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in c_n around 0 88.9%

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

   6. Taylor expanded in s around -inf 88.9%

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

   8. Simplified 97.4%

      \[\leadsto \color{blue}{{\left(e^{c\_p}\right)}^{\left(\left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 97.4%

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

      2. exp-prod 97.4%

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

   10. Applied egg-rr 97.4%

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

       1. exp-1-e 97.4%

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

   12. Simplified 97.4%

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

   13. Taylor expanded in s around 0 98.6%

       \[\leadsto {\left({e}^{c\_p}\right)}^{\color{blue}{\left(\left(\log \left(1 + e^{-t}\right) + s \cdot \left(0.5 + -0.125 \cdot s\right)\right) - \log 2\right)}} \]
   14. Step-by-step derivation

       1. associate--l+ 98.6%

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

       2. log1p-define 98.6%

          \[\leadsto {\left({e}^{c\_p}\right)}^{\left(\color{blue}{\mathsf{log1p}\left(e^{-t}\right)} + \left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right)\right)} \]

       3. *-commutative 98.6%

          \[\leadsto {\left({e}^{c\_p}\right)}^{\left(\mathsf{log1p}\left(e^{-t}\right) + \left(s \cdot \left(0.5 + \color{blue}{s \cdot -0.125}\right) - \log 2\right)\right)} \]

   15. Simplified 98.6%

       \[\leadsto {\left({e}^{c\_p}\right)}^{\color{blue}{\left(\mathsf{log1p}\left(e^{-t}\right) + \left(s \cdot \left(0.5 + s \cdot -0.125\right) - \log 2\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{e^{\mathsf{fma}\left(\mathsf{log1p}\left(\frac{1}{1 + e^{s}}\right), c\_n, \mathsf{log1p}\left(e^{s}\right) \cdot \left(-c\_p\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right)}}{{\left(1 + e^{t}\right)}^{\left(-c\_p\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left({e}^{c\_p}\right)}^{\left(\mathsf{log1p}\left(e^{-t}\right) + \left(s \cdot \left(0.5 + s \cdot -0.125\right) - \log 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-t}\\ \mathbf{if}\;-t \leq 2 \cdot 10^{-18}:\\ \;\;\;\;e^{\left(s \cdot \left(-0.125 \cdot \left(c\_p \cdot s\right) + c\_p \cdot 0.5\right) - c\_p \cdot \log 2\right) + c\_p \cdot \mathsf{log1p}\left(t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - t\_1}\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- t))))
   (if (<= (- t) 2e-18)
     (exp
      (+
       (- (* s (+ (* -0.125 (* c_p s)) (* c_p 0.5))) (* c_p (log 2.0)))
       (* c_p (log1p t_1))))
     (/
      (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-18) {
		tmp = exp((((s * ((-0.125 * (c_p * s)) + (c_p * 0.5))) - (c_p * log(2.0))) + (c_p * log1p(t_1))));
	} 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;
}

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-18) {
		tmp = Math.exp((((s * ((-0.125 * (c_p * s)) + (c_p * 0.5))) - (c_p * Math.log(2.0))) + (c_p * Math.log1p(t_1))));
	} 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-18:
		tmp = math.exp((((s * ((-0.125 * (c_p * s)) + (c_p * 0.5))) - (c_p * math.log(2.0))) + (c_p * math.log1p(t_1))))
	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-18)
		tmp = exp(Float64(Float64(Float64(s * Float64(Float64(-0.125 * Float64(c_p * s)) + Float64(c_p * 0.5))) - Float64(c_p * log(2.0))) + Float64(c_p * log1p(t_1))));
	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

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-t)], $MachinePrecision]}, If[LessEqual[(-t), 2e-18], N[Exp[N[(N[(N[(s * N[(N[(-0.125 * N[(c$95$p * s), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c$95$p * N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision]), $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) < 2.0000000000000001e-18

   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 95.1%

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

      1. add-exp-log 95.1%

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

      2. log-div 95.1%

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

      3. log-pow 95.1%

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

      4. log-rec 95.1%

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

      5. log1p-define 95.1%

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

      6. log-pow 97.1%

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

      7. neg-log 97.1%

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

      8. log1p-define 97.1%

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

   7. Applied egg-rr 97.1%

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

   8. Taylor expanded in s around 0 99.4%

      \[\leadsto e^{\color{blue}{\left(-1 \cdot \left(c\_p \cdot \log 2\right) + s \cdot \left(-0.125 \cdot \left(c\_p \cdot s\right) + 0.5 \cdot c\_p\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   if 2.0000000000000001e-18 < (neg.f64 t)

   1. Initial program 34.0%

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

      1. associate-/l/ 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in c_p around 0 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 2 \cdot 10^{-18}:\\ \;\;\;\;e^{\left(s \cdot \left(-0.125 \cdot \left(c\_p \cdot s\right) + c\_p \cdot 0.5\right) - c\_p \cdot \log 2\right) + c\_p \cdot \mathsf{log1p}\left(e^{-t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-t}\\ \mathbf{if}\;-t \leq 2 \cdot 10^{-18}:\\ \;\;\;\;e^{c\_p \cdot \left(s \cdot \left(0.5 + s \cdot -0.125\right) - \log 2\right) + c\_p \cdot \mathsf{log1p}\left(t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - t\_1}\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- t))))
   (if (<= (- t) 2e-18)
     (exp
      (+ (* c_p (- (* s (+ 0.5 (* s -0.125))) (log 2.0))) (* c_p (log1p t_1))))
     (/
      (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-18) {
		tmp = exp(((c_p * ((s * (0.5 + (s * -0.125))) - log(2.0))) + (c_p * log1p(t_1))));
	} 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;
}

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-18) {
		tmp = Math.exp(((c_p * ((s * (0.5 + (s * -0.125))) - Math.log(2.0))) + (c_p * Math.log1p(t_1))));
	} 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-18:
		tmp = math.exp(((c_p * ((s * (0.5 + (s * -0.125))) - math.log(2.0))) + (c_p * math.log1p(t_1))))
	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-18)
		tmp = exp(Float64(Float64(c_p * Float64(Float64(s * Float64(0.5 + Float64(s * -0.125))) - log(2.0))) + Float64(c_p * log1p(t_1))));
	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

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-t)], $MachinePrecision]}, If[LessEqual[(-t), 2e-18], N[Exp[N[(N[(c$95$p * N[(N[(s * N[(0.5 + N[(s * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision]), $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) < 2.0000000000000001e-18

   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 95.1%

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

      1. add-exp-log 95.1%

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

      2. log-div 95.1%

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

      3. log-pow 95.1%

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

      4. log-rec 95.1%

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

      5. log1p-define 95.1%

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

      6. log-pow 97.1%

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

      7. neg-log 97.1%

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

      8. log1p-define 97.1%

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

   7. Applied egg-rr 97.1%

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

   8. Taylor expanded in s around 0 99.4%

      \[\leadsto e^{c\_p \cdot \color{blue}{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto e^{c\_p \cdot \left(s \cdot \left(0.5 + \color{blue}{s \cdot -0.125}\right) - \log 2\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   10. Simplified 99.4%

       \[\leadsto e^{c\_p \cdot \color{blue}{\left(s \cdot \left(0.5 + s \cdot -0.125\right) - \log 2\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   if 2.0000000000000001e-18 < (neg.f64 t)

   1. Initial program 34.0%

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

      1. associate-/l/ 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in c_p around 0 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 2 \cdot 10^{-18}:\\ \;\;\;\;e^{c\_p \cdot \left(s \cdot \left(0.5 + s \cdot -0.125\right) - \log 2\right) + c\_p \cdot \mathsf{log1p}\left(e^{-t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(1 + \frac{1}{-1 - e^{-s}}\right)}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-t \leq 2 \cdot 10^{-18}:\\ \;\;\;\;e^{c\_p \cdot \left(s \cdot \left(0.5 + s \cdot -0.125\right) - \log 2\right) + c\_p \cdot \mathsf{log1p}\left(e^{-t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- t) 2e-18)
   (exp
    (+
     (* c_p (- (* s (+ 0.5 (* s -0.125))) (log 2.0)))
     (* c_p (log1p (exp (- t))))))
   (/ (pow 0.5 c_n) (pow (+ 0.5 (* t -0.25)) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= 2e-18) {
		tmp = exp(((c_p * ((s * (0.5 + (s * -0.125))) - log(2.0))) + (c_p * log1p(exp(-t)))));
	} else {
		tmp = pow(0.5, c_n) / pow((0.5 + (t * -0.25)), c_n);
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= 2e-18) {
		tmp = Math.exp(((c_p * ((s * (0.5 + (s * -0.125))) - Math.log(2.0))) + (c_p * Math.log1p(Math.exp(-t)))));
	} else {
		tmp = Math.pow(0.5, c_n) / Math.pow((0.5 + (t * -0.25)), c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -t <= 2e-18:
		tmp = math.exp(((c_p * ((s * (0.5 + (s * -0.125))) - math.log(2.0))) + (c_p * math.log1p(math.exp(-t)))))
	else:
		tmp = math.pow(0.5, c_n) / math.pow((0.5 + (t * -0.25)), c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-t) <= 2e-18)
		tmp = exp(Float64(Float64(c_p * Float64(Float64(s * Float64(0.5 + Float64(s * -0.125))) - log(2.0))) + Float64(c_p * log1p(exp(Float64(-t))))));
	else
		tmp = Float64((0.5 ^ c_n) / (Float64(0.5 + Float64(t * -0.25)) ^ c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-t), 2e-18], N[Exp[N[(N[(c$95$p * N[(N[(s * N[(0.5 + N[(s * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 2.0000000000000001e-18

   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 95.1%

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

      1. add-exp-log 95.1%

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

      2. log-div 95.1%

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

      3. log-pow 95.1%

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

      4. log-rec 95.1%

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

      5. log1p-define 95.1%

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

      6. log-pow 97.1%

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

      7. neg-log 97.1%

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

      8. log1p-define 97.1%

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

   7. Applied egg-rr 97.1%

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

   8. Taylor expanded in s around 0 99.4%

      \[\leadsto e^{c\_p \cdot \color{blue}{\left(s \cdot \left(0.5 + -0.125 \cdot s\right) - \log 2\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto e^{c\_p \cdot \left(s \cdot \left(0.5 + \color{blue}{s \cdot -0.125}\right) - \log 2\right) - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   10. Simplified 99.4%

       \[\leadsto e^{c\_p \cdot \color{blue}{\left(s \cdot \left(0.5 + s \cdot -0.125\right) - \log 2\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   if 2.0000000000000001e-18 < (neg.f64 t)

   1. Initial program 34.0%

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

      1. associate-/l/ 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in c_p around 0 100.0%

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

   6. Taylor expanded in s around 0 89.2%

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

   7. Taylor expanded in t around 0 89.2%

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

      1. *-commutative 89.2%

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

   9. Simplified 89.2%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\color{blue}{\left(0.5 + t \cdot -0.25\right)}}^{c\_n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 2 \cdot 10^{-18}:\\ \;\;\;\;e^{c\_p \cdot \left(s \cdot \left(0.5 + s \cdot -0.125\right) - \log 2\right) + c\_p \cdot \mathsf{log1p}\left(e^{-t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 0.00033:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot 0.5\right)}\right)}^{c\_p}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;e^{s \cdot \left(c\_p \cdot 0.5\right) + c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \log 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 0.00033)
   (/ (pow (/ 1.0 (+ 2.0 (* s (+ -1.0 (* s 0.5))))) c_p) (pow 0.5 c_p))
   (exp (+ (* s (* c_p 0.5)) (* c_p (- (log1p (exp (- t))) (log 2.0)))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 0.00033) {
		tmp = pow((1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))), c_p) / pow(0.5, c_p);
	} else {
		tmp = exp(((s * (c_p * 0.5)) + (c_p * (log1p(exp(-t)) - log(2.0)))));
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 0.00033) {
		tmp = Math.pow((1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))), c_p) / Math.pow(0.5, c_p);
	} else {
		tmp = Math.exp(((s * (c_p * 0.5)) + (c_p * (Math.log1p(Math.exp(-t)) - Math.log(2.0)))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 0.00033:
		tmp = math.pow((1.0 / (2.0 + (s * (-1.0 + (s * 0.5))))), c_p) / math.pow(0.5, c_p)
	else:
		tmp = math.exp(((s * (c_p * 0.5)) + (c_p * (math.log1p(math.exp(-t)) - math.log(2.0)))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 0.00033)
		tmp = Float64((Float64(1.0 / Float64(2.0 + Float64(s * Float64(-1.0 + Float64(s * 0.5))))) ^ c_p) / (0.5 ^ c_p));
	else
		tmp = exp(Float64(Float64(s * Float64(c_p * 0.5)) + Float64(c_p * Float64(log1p(exp(Float64(-t))) - log(2.0)))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 0.00033], N[(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] / N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(s * N[(c$95$p * 0.5), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * N[(N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 3.3e-4

   1. Initial program 93.0%

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

      1. associate-/l/ 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in c_n around 0 94.6%

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

   6. Taylor expanded in s around 0 97.0%

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

   7. Taylor expanded in t around 0 98.7%

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

   if 3.3e-4 < c_p

   1. Initial program 64.2%

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

      1. associate-/l/ 64.2%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in c_n around 0 70.1%

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

      1. add-exp-log 70.1%

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

      2. log-div 70.1%

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

      3. log-pow 70.6%

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

      4. log-rec 70.6%

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

      5. log1p-define 70.6%

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

      6. log-pow 98.8%

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

      7. neg-log 98.8%

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

      8. log1p-define 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in s around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. associate--l+ 98.6%

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

      3. *-commutative 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-*r* 98.6%

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

      6. sub-neg 98.6%

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

      7. mul-1-neg 98.6%

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

      8. remove-double-neg 98.6%

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

      9. cancel-sign-sub 98.6%

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

      10. associate-*r* 98.6%

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

      11. neg-mul-1 98.6%

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

      12. distribute-lft-out-- 98.6%

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

      13. log1p-define 98.6%

          \[\leadsto e^{s \cdot \left(c\_p \cdot 0.5\right) + \left(-c\_p\right) \cdot \left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-t}\right)}\right)} \]

   10. Simplified 98.6%

       \[\leadsto e^{\color{blue}{s \cdot \left(c\_p \cdot 0.5\right) + \left(-c\_p\right) \cdot \left(\log 2 - \mathsf{log1p}\left(e^{-t}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 0.00033:\\ \;\;\;\;\frac{{\left(\frac{1}{2 + s \cdot \left(-1 + s \cdot 0.5\right)}\right)}^{c\_p}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;e^{s \cdot \left(c\_p \cdot 0.5\right) + c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \log 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

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 = exp(((s * (c_p * 0.5d0)) + ((-0.5d0) * (c_p * t))))
end function

Java

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

Python

def code(c_p, c_n, t, s):
	return math.exp(((s * (c_p * 0.5)) + (-0.5 * (c_p * t))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-/l/ 91.1%

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

3. Simplified 91.1%

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

5. Taylor expanded in c_n around 0 93.0%

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

   1. add-exp-log 93.0%

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

   2. log-div 93.0%

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

   3. log-pow 93.0%

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

   4. log-rec 93.0%

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

   5. log1p-define 93.0%

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

   6. log-pow 94.9%

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

   7. neg-log 94.9%

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

   8. log1p-define 94.9%

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

7. Applied egg-rr 94.9%

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

8. Taylor expanded in s around 0 95.9%

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

   1. +-commutative 95.9%

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

   2. associate--l+ 96.0%

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

   3. *-commutative 96.0%

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

   4. *-commutative 96.0%

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

   5. associate-*r* 96.0%

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

   6. sub-neg 96.0%

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

   7. mul-1-neg 96.0%

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

   8. remove-double-neg 96.0%

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

   9. cancel-sign-sub 96.0%

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

   10. associate-*r* 96.0%

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

   11. neg-mul-1 96.0%

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

   12. distribute-lft-out-- 96.0%

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

   13. log1p-define 96.0%

       \[\leadsto e^{s \cdot \left(c\_p \cdot 0.5\right) + \left(-c\_p\right) \cdot \left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-t}\right)}\right)} \]

10. Simplified 96.0%

    \[\leadsto e^{\color{blue}{s \cdot \left(c\_p \cdot 0.5\right) + \left(-c\_p\right) \cdot \left(\log 2 - \mathsf{log1p}\left(e^{-t}\right)\right)}} \]

11. Taylor expanded in t around 0 96.9%

    \[\leadsto e^{s \cdot \left(c\_p \cdot 0.5\right) + \color{blue}{-0.5 \cdot \left(c\_p \cdot t\right)}} \]
12. Add Preprocessing

Alternative 7: 94.3% accurate, 835.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-/l/ 91.1%

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

3. Simplified 91.1%

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

5. Taylor expanded in c_n around 0 93.0%

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

6. Taylor expanded in c_p around 0 93.1%

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

Developer Target 1: 96.3% 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 2024145 
(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))))