Harley's example

Percentage Accurate: 91.0% → 97.5%

Time: 52.9s

Alternatives: 9

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 8.674421414310918e-42
  2. c_n = 2.8257158229687884e-272
  3. t = 7.448034528831689e+79
  4. s = 4.172938683981005e+124

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s}\\ t_2 := e^{-t}\\ t_3 := \frac{-1}{t\_2 + 1}\\ \mathbf{if}\;-t \leq 5 \cdot 10^{-162}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(t\_2\right) - \mathsf{log1p}\left(t\_1\right), c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 - t\_1}\right) - \mathsf{log1p}\left(t\_3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{1 + t\_3}{1 + \frac{-1}{t\_1 + 1}}\right) \cdot \left(-c\_n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- s))) (t_2 (exp (- t))) (t_3 (/ -1.0 (+ t_2 1.0))))
   (if (<= (- t) 5e-162)
     (exp
      (fma
       c_p
       (- (log1p t_2) (log1p t_1))
       (* c_n (- (log1p (/ 1.0 (- -1.0 t_1))) (log1p t_3)))))
     (exp (* (log (/ (+ 1.0 t_3) (+ 1.0 (/ -1.0 (+ t_1 1.0))))) (- c_n))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-s);
	double t_2 = exp(-t);
	double t_3 = -1.0 / (t_2 + 1.0);
	double tmp;
	if (-t <= 5e-162) {
		tmp = exp(fma(c_p, (log1p(t_2) - log1p(t_1)), (c_n * (log1p((1.0 / (-1.0 - t_1))) - log1p(t_3)))));
	} else {
		tmp = exp((log(((1.0 + t_3) / (1.0 + (-1.0 / (t_1 + 1.0))))) * -c_n));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-s))
	t_2 = exp(Float64(-t))
	t_3 = Float64(-1.0 / Float64(t_2 + 1.0))
	tmp = 0.0
	if (Float64(-t) <= 5e-162)
		tmp = exp(fma(c_p, Float64(log1p(t_2) - log1p(t_1)), Float64(c_n * Float64(log1p(Float64(1.0 / Float64(-1.0 - t_1))) - log1p(t_3)))));
	else
		tmp = exp(Float64(log(Float64(Float64(1.0 + t_3) / Float64(1.0 + Float64(-1.0 / Float64(t_1 + 1.0))))) * Float64(-c_n)));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-s)], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t)], $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 / N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[(-t), 5e-162], N[Exp[N[(c$95$p * N[(N[Log[1 + t$95$2], $MachinePrecision] - N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision] + N[(c$95$n * N[(N[Log[1 + N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Log[N[(N[(1.0 + t$95$3), $MachinePrecision] / N[(1.0 + N[(-1.0 / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-c$95$n)), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 5.00000000000000014e-162

   1. Initial program 94.1%

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

   4. Applied rewrites 100.0%

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

   if 5.00000000000000014e-162 < (neg.f64 t)

   1. Initial program 82.6%

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in c_p around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 99.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. diff-log N/A

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

   9. Applied rewrites 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 5 \cdot 10^{-162}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right), c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 - e^{-s}}\right) - \mathsf{log1p}\left(\frac{-1}{e^{-t} + 1}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{1 + \frac{-1}{e^{-t} + 1}}{1 + \frac{-1}{e^{-s} + 1}}\right) \cdot \left(-c\_n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 5 \cdot 10^{-92}:\\ \;\;\;\;e^{\log \left(\frac{1 + \frac{-1}{e^{-t} + 1}}{1 + \frac{-1}{e^{-s} + 1}}\right) \cdot \left(-c\_n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(s \cdot 0.5, \mathsf{fma}\left(s, 0.5, -1\right), 1\right)\right)}^{\left(-c\_p\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 5e-92)
   (exp
    (*
     (log
      (/
       (+ 1.0 (/ -1.0 (+ (exp (- t)) 1.0)))
       (+ 1.0 (/ -1.0 (+ (exp (- s)) 1.0)))))
     (- c_n)))
   (pow (fma (* s 0.5) (fma s 0.5 -1.0) 1.0) (- c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 5e-92) {
		tmp = exp((log(((1.0 + (-1.0 / (exp(-t) + 1.0))) / (1.0 + (-1.0 / (exp(-s) + 1.0))))) * -c_n));
	} else {
		tmp = pow(fma((s * 0.5), fma(s, 0.5, -1.0), 1.0), -c_p);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 5e-92)
		tmp = exp(Float64(log(Float64(Float64(1.0 + Float64(-1.0 / Float64(exp(Float64(-t)) + 1.0))) / Float64(1.0 + Float64(-1.0 / Float64(exp(Float64(-s)) + 1.0))))) * Float64(-c_n)));
	else
		tmp = fma(Float64(s * 0.5), fma(s, 0.5, -1.0), 1.0) ^ Float64(-c_p);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 5e-92], N[Exp[N[(N[Log[N[(N[(1.0 + N[(-1.0 / N[(N[Exp[(-t)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 / N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-c$95$n)), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(s * 0.5), $MachinePrecision] * N[(s * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], (-c$95$p)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 5.00000000000000011e-92

   1. Initial program 91.5%

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in c_p around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 99.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. diff-log N/A

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

   9. Applied rewrites 99.8%

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

   if 5.00000000000000011e-92 < c_p

   1. Initial program 90.2%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in s around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{2} \cdot s + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. *-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{s \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. metadata-eval N/A

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

      6. lower-fma.f64 90.2

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

   8. Applied rewrites 90.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \frac{1}{2}, -1\right), 2\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \frac{1}{2}, -1\right), 2\right)}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. *-lft-identity N/A

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

      5. unpow-prod-down N/A

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

      6. pow-base-1 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. rem-exp-log N/A

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

      13. rem-exp-log N/A

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

      14. *-lft-identity N/A

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

      15. unpow-prod-down N/A

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. sub-neg N/A

          \[\leadsto {\left(\mathsf{fma}\left(s \cdot \frac{1}{2}, \color{blue}{\frac{1}{2} \cdot s + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)\right)}^{\left(\mathsf{neg}\left(c\_p\right)\right)} \]

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 100.0

           \[\leadsto {\left(\mathsf{fma}\left(s \cdot 0.5, \color{blue}{\mathsf{fma}\left(s, 0.5, -1\right)}, 1\right)\right)}^{\left(-c\_p\right)} \]

   13. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 5 \cdot 10^{-92}:\\ \;\;\;\;e^{\log \left(\frac{1 + \frac{-1}{e^{-t} + 1}}{1 + \frac{-1}{e^{-s} + 1}}\right) \cdot \left(-c\_n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(s \cdot 0.5, \mathsf{fma}\left(s, 0.5, -1\right), 1\right)\right)}^{\left(-c\_p\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 5 \cdot 10^{-92}:\\ \;\;\;\;{\left(\frac{1 + \frac{-1}{e^{-s} + 1}}{1 + \frac{-1}{e^{-t} + 1}}\right)}^{c\_n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(s \cdot 0.5, \mathsf{fma}\left(s, 0.5, -1\right), 1\right)\right)}^{\left(-c\_p\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 5e-92)
   (pow
    (/
     (+ 1.0 (/ -1.0 (+ (exp (- s)) 1.0)))
     (+ 1.0 (/ -1.0 (+ (exp (- t)) 1.0))))
    c_n)
   (pow (fma (* s 0.5) (fma s 0.5 -1.0) 1.0) (- c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 5e-92) {
		tmp = pow(((1.0 + (-1.0 / (exp(-s) + 1.0))) / (1.0 + (-1.0 / (exp(-t) + 1.0)))), c_n);
	} else {
		tmp = pow(fma((s * 0.5), fma(s, 0.5, -1.0), 1.0), -c_p);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 5e-92)
		tmp = Float64(Float64(1.0 + Float64(-1.0 / Float64(exp(Float64(-s)) + 1.0))) / Float64(1.0 + Float64(-1.0 / Float64(exp(Float64(-t)) + 1.0)))) ^ c_n;
	else
		tmp = fma(Float64(s * 0.5), fma(s, 0.5, -1.0), 1.0) ^ Float64(-c_p);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 5e-92], N[Power[N[(N[(1.0 + N[(-1.0 / N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 / N[(N[Exp[(-t)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision], N[Power[N[(N[(s * 0.5), $MachinePrecision] * N[(s * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], (-c$95$p)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 5.00000000000000011e-92

   1. Initial program 91.5%

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in c_p around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 99.8%

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

   if 5.00000000000000011e-92 < c_p

   1. Initial program 90.2%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in s around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{2} \cdot s + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. *-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{s \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. metadata-eval N/A

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

      6. lower-fma.f64 90.2

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

   8. Applied rewrites 90.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \frac{1}{2}, -1\right), 2\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \frac{1}{2}, -1\right), 2\right)}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. *-lft-identity N/A

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

      5. unpow-prod-down N/A

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

      6. pow-base-1 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. rem-exp-log N/A

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

      13. rem-exp-log N/A

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

      14. *-lft-identity N/A

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

      15. unpow-prod-down N/A

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. sub-neg N/A

          \[\leadsto {\left(\mathsf{fma}\left(s \cdot \frac{1}{2}, \color{blue}{\frac{1}{2} \cdot s + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)\right)}^{\left(\mathsf{neg}\left(c\_p\right)\right)} \]

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 100.0

           \[\leadsto {\left(\mathsf{fma}\left(s \cdot 0.5, \color{blue}{\mathsf{fma}\left(s, 0.5, -1\right)}, 1\right)\right)}^{\left(-c\_p\right)} \]

   13. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 5 \cdot 10^{-92}:\\ \;\;\;\;{\left(\frac{1 + \frac{-1}{e^{-s} + 1}}{1 + \frac{-1}{e^{-t} + 1}}\right)}^{c\_n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(s \cdot 0.5, \mathsf{fma}\left(s, 0.5, -1\right), 1\right)\right)}^{\left(-c\_p\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-t \leq 0.0005:\\ \;\;\;\;{\left(\mathsf{fma}\left(s \cdot 0.5, \mathsf{fma}\left(s, 0.5, -1\right), 1\right)\right)}^{\left(-c\_p\right)}\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_n}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- t) 0.0005)
   (pow (fma (* s 0.5) (fma s 0.5 -1.0) 1.0) (- c_p))
   (pow 0.5 c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= 0.0005) {
		tmp = pow(fma((s * 0.5), fma(s, 0.5, -1.0), 1.0), -c_p);
	} else {
		tmp = pow(0.5, c_n);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-t) <= 0.0005)
		tmp = fma(Float64(s * 0.5), fma(s, 0.5, -1.0), 1.0) ^ Float64(-c_p);
	else
		tmp = 0.5 ^ c_n;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-t), 0.0005], N[Power[N[(N[(s * 0.5), $MachinePrecision] * N[(s * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], (-c$95$p)], $MachinePrecision], N[Power[0.5, c$95$n], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 5.0000000000000001e-4

   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. Add Preprocessing

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 95.5

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in s around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{2} \cdot s + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. *-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{s \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. metadata-eval N/A

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

      6. lower-fma.f64 95.9

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

   8. Applied rewrites 95.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \frac{{\left(\frac{1}{\color{blue}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \frac{1}{2}, -1\right), 2\right)}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \frac{1}{2}, -1\right), 2\right)}\right)}}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. *-lft-identity N/A

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

      5. unpow-prod-down N/A

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

      6. pow-base-1 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. rem-exp-log N/A

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

      13. rem-exp-log N/A

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

      14. *-lft-identity N/A

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

      15. unpow-prod-down N/A

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. sub-neg N/A

          \[\leadsto {\left(\mathsf{fma}\left(s \cdot \frac{1}{2}, \color{blue}{\frac{1}{2} \cdot s + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)\right)}^{\left(\mathsf{neg}\left(c\_p\right)\right)} \]

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lower-fma.f64 99.5

           \[\leadsto {\left(\mathsf{fma}\left(s \cdot 0.5, \color{blue}{\mathsf{fma}\left(s, 0.5, -1\right)}, 1\right)\right)}^{\left(-c\_p\right)} \]

   13. Applied rewrites 99.5%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(s \cdot 0.5, \mathsf{fma}\left(s, 0.5, -1\right), 1\right)\right)}}^{\left(-c\_p\right)} \]

   if 5.0000000000000001e-4 < (neg.f64 t)

   1. Initial program 45.5%

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

   3. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-pow.f64 N/A

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

   5. Applied rewrites 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

      \[\leadsto \color{blue}{\frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\frac{1}{2}}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      4. sub-neg N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)\right)}}^{c\_n}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)\right)}}^{c\_n}} \]

      6. distribute-neg-frac N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(t\right)}}}\right)}^{c\_n}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 + \frac{\color{blue}{-1}}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 + \color{blue}{\frac{-1}{1 + e^{\mathsf{neg}\left(t\right)}}}\right)}^{c\_n}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 + \frac{-1}{\color{blue}{1 + e^{\mathsf{neg}\left(t\right)}}}\right)}^{c\_n}} \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 + \frac{-1}{1 + \color{blue}{e^{\mathsf{neg}\left(t\right)}}}\right)}^{c\_n}} \]

      11. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in c_n around 0

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

   11. Applied rewrites 100.0%

       \[\leadsto \frac{{0.5}^{c\_n}}{\color{blue}{1}} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{\frac{1}{2}}^{c\_n}}}{1} \]

       2. /-rgt-identity 100.0

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

   13. Applied rewrites 100.0%

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

Alternative 5: 95.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \mathsf{fma}\left(c\_n, 0.125, -0.125\right), c\_n \cdot -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(s, \mathsf{fma}\left(s, 0.5, -1\right), 2\right)\right)}^{\left(-c\_p\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 0.02)
   (fma s (fma s (* c_n (fma c_n 0.125 -0.125)) (* c_n -0.5)) 1.0)
   (pow (fma s (fma s 0.5 -1.0) 2.0) (- c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 0.02) {
		tmp = fma(s, fma(s, (c_n * fma(c_n, 0.125, -0.125)), (c_n * -0.5)), 1.0);
	} else {
		tmp = pow(fma(s, fma(s, 0.5, -1.0), 2.0), -c_p);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 0.02)
		tmp = fma(s, fma(s, Float64(c_n * fma(c_n, 0.125, -0.125)), Float64(c_n * -0.5)), 1.0);
	else
		tmp = fma(s, fma(s, 0.5, -1.0), 2.0) ^ Float64(-c_p);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 0.02], N[(s * N[(s * N[(c$95$n * N[(c$95$n * 0.125 + -0.125), $MachinePrecision]), $MachinePrecision] + N[(c$95$n * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[Power[N[(s * N[(s * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 0.0200000000000000004

   1. Initial program 91.5%

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

   3. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-pow.f64 N/A

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\frac{1}{2}}^{c\_n}} \]

      3. sub-neg N/A

         \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\color{blue}{-1 \cdot s}}}\right)\right)\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      6. distribute-neg-frac N/A

         \[\leadsto \frac{{\left(1 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{-1 \cdot s}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. neg-mul-1 N/A

          \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      12. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      13. lower-pow.f64 93.9

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

   8. Applied rewrites 93.9%

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

   9. Taylor expanded in s around 0

      \[\leadsto \color{blue}{1 + s \cdot \left(\frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{s \cdot \left(\frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right)\right) + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(s, \frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right), 1\right)} \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(s, \color{blue}{s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right) + \frac{-1}{2} \cdot c\_n}, 1\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(s, \frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{\frac{1}{8} \cdot {c\_n}^{2} + \frac{-1}{8} \cdot c\_n}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{{c\_n}^{2} \cdot \frac{1}{8}} + \frac{-1}{8} \cdot c\_n, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{\left(c\_n \cdot c\_n\right)} \cdot \frac{1}{8} + \frac{-1}{8} \cdot c\_n, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{c\_n \cdot \left(c\_n \cdot \frac{1}{8}\right)} + \frac{-1}{8} \cdot c\_n, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \left(c\_n \cdot \frac{1}{8}\right) + \color{blue}{c\_n \cdot \frac{-1}{8}}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       10. distribute-lft-out N/A

           \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{c\_n \cdot \left(c\_n \cdot \frac{1}{8} + \frac{-1}{8}\right)}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{c\_n \cdot \left(c\_n \cdot \frac{1}{8} + \frac{-1}{8}\right)}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       12. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \color{blue}{\mathsf{fma}\left(c\_n, \frac{1}{8}, \frac{-1}{8}\right)}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       13. lower-*.f64 96.4

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

   11. Applied rewrites 96.4%

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

   if 0.0200000000000000004 < (neg.f64 s)

   1. Initial program 77.8%

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

   3. Taylor expanded in c_n around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in s around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{\frac{1}{2} \cdot s + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      4. *-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{s \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}} \]

      5. metadata-eval N/A

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

      6. lower-fma.f64 77.8

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

   8. Applied rewrites 77.8%

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

   9. Taylor expanded in c_p around 0

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \frac{1}{2}, -1\right), 2\right)}\right)}^{c\_p}}{\color{blue}{1}} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 0.5, -1\right), 2\right)}\right)}^{c\_p}}{\color{blue}{1}} \]
   12. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. /-rgt-identity 100.0

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

       6. lift-pow.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. inv-pow N/A

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

       9. pow-pow N/A

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

       10. neg-mul-1 N/A

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

       11. lift-neg.f64 N/A

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

       12. lower-pow.f64 100.0

           \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(s, \mathsf{fma}\left(s, 0.5, -1\right), 2\right)\right)}^{\left(-c\_p\right)}} \]

   13. Applied rewrites 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \mathsf{fma}\left(c\_n, 0.125, -0.125\right), c\_n \cdot -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(s, \mathsf{fma}\left(s, 0.5, -1\right), 2\right)\right)}^{\left(-c\_p\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-t \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \mathsf{fma}\left(c\_n, 0.125, -0.125\right), c\_n \cdot -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_n}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- t) 0.0005)
   (fma s (fma s (* c_n (fma c_n 0.125 -0.125)) (* c_n -0.5)) 1.0)
   (pow 0.5 c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= 0.0005) {
		tmp = fma(s, fma(s, (c_n * fma(c_n, 0.125, -0.125)), (c_n * -0.5)), 1.0);
	} else {
		tmp = pow(0.5, c_n);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-t) <= 0.0005)
		tmp = fma(s, fma(s, Float64(c_n * fma(c_n, 0.125, -0.125)), Float64(c_n * -0.5)), 1.0);
	else
		tmp = 0.5 ^ c_n;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-t), 0.0005], N[(s * N[(s * N[(c$95$n * N[(c$95$n * 0.125 + -0.125), $MachinePrecision]), $MachinePrecision] + N[(c$95$n * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[Power[0.5, c$95$n], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < 5.0000000000000001e-4

   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. Add Preprocessing

   3. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-pow.f64 N/A

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\frac{1}{2}}^{c\_n}} \]

      3. sub-neg N/A

         \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      5. neg-mul-1 N/A

         \[\leadsto \frac{{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\color{blue}{-1 \cdot s}}}\right)\right)\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      6. distribute-neg-frac N/A

         \[\leadsto \frac{{\left(1 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{-1 \cdot s}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. neg-mul-1 N/A

          \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      12. lower-neg.f64 N/A

          \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

      13. lower-pow.f64 93.9

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

   8. Applied rewrites 93.9%

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

   9. Taylor expanded in s around 0

      \[\leadsto \color{blue}{1 + s \cdot \left(\frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{s \cdot \left(\frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right)\right) + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(s, \frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right), 1\right)} \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(s, \color{blue}{s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right) + \frac{-1}{2} \cdot c\_n}, 1\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(s, \frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{\frac{1}{8} \cdot {c\_n}^{2} + \frac{-1}{8} \cdot c\_n}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{{c\_n}^{2} \cdot \frac{1}{8}} + \frac{-1}{8} \cdot c\_n, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{\left(c\_n \cdot c\_n\right)} \cdot \frac{1}{8} + \frac{-1}{8} \cdot c\_n, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{c\_n \cdot \left(c\_n \cdot \frac{1}{8}\right)} + \frac{-1}{8} \cdot c\_n, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \left(c\_n \cdot \frac{1}{8}\right) + \color{blue}{c\_n \cdot \frac{-1}{8}}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       10. distribute-lft-out N/A

           \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{c\_n \cdot \left(c\_n \cdot \frac{1}{8} + \frac{-1}{8}\right)}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{c\_n \cdot \left(c\_n \cdot \frac{1}{8} + \frac{-1}{8}\right)}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       12. lower-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \color{blue}{\mathsf{fma}\left(c\_n, \frac{1}{8}, \frac{-1}{8}\right)}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

       13. lower-*.f64 96.4

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

   11. Applied rewrites 96.4%

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

   if 5.0000000000000001e-4 < (neg.f64 t)

   1. Initial program 45.5%

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

   3. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-+.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-pow.f64 N/A

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

   5. Applied rewrites 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

      \[\leadsto \color{blue}{\frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{\frac{1}{2}}^{c\_n}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\frac{1}{2}}^{c\_n}}}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}}} \]

      4. sub-neg N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)\right)}}^{c\_n}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)\right)\right)}}^{c\_n}} \]

      6. distribute-neg-frac N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{\mathsf{neg}\left(t\right)}}}\right)}^{c\_n}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 + \frac{\color{blue}{-1}}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_n}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 + \color{blue}{\frac{-1}{1 + e^{\mathsf{neg}\left(t\right)}}}\right)}^{c\_n}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 + \frac{-1}{\color{blue}{1 + e^{\mathsf{neg}\left(t\right)}}}\right)}^{c\_n}} \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(1 + \frac{-1}{1 + \color{blue}{e^{\mathsf{neg}\left(t\right)}}}\right)}^{c\_n}} \]

      11. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in c_n around 0

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

   11. Applied rewrites 100.0%

       \[\leadsto \frac{{0.5}^{c\_n}}{\color{blue}{1}} \]
   12. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto \frac{\color{blue}{{\frac{1}{2}}^{c\_n}}}{1} \]

       2. /-rgt-identity 100.0

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

   13. Applied rewrites 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \mathsf{fma}\left(c\_n, 0.125, -0.125\right), c\_n \cdot -0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.1% accurate, 30.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in c_p around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-+.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-pow.f64 N/A

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

5. Applied rewrites 94.2%

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

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\frac{1}{2}}^{c\_n}} \]

   3. sub-neg N/A

      \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   5. neg-mul-1 N/A

      \[\leadsto \frac{{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\color{blue}{-1 \cdot s}}}\right)\right)\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   6. distribute-neg-frac N/A

      \[\leadsto \frac{{\left(1 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{-1 \cdot s}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. neg-mul-1 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   12. lower-neg.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   13. lower-pow.f64 91.8

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

8. Applied rewrites 91.8%

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

9. Taylor expanded in s around 0

   \[\leadsto \color{blue}{1 + s \cdot \left(\frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{s \cdot \left(\frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right)\right) + 1} \]

    2. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(s, \frac{-1}{2} \cdot c\_n + s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right), 1\right)} \]

    3. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(s, \color{blue}{s \cdot \left(\frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}\right) + \frac{-1}{2} \cdot c\_n}, 1\right) \]

    4. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(s, \frac{-1}{8} \cdot c\_n + \frac{1}{8} \cdot {c\_n}^{2}, \frac{-1}{2} \cdot c\_n\right)}, 1\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{\frac{1}{8} \cdot {c\_n}^{2} + \frac{-1}{8} \cdot c\_n}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{{c\_n}^{2} \cdot \frac{1}{8}} + \frac{-1}{8} \cdot c\_n, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{\left(c\_n \cdot c\_n\right)} \cdot \frac{1}{8} + \frac{-1}{8} \cdot c\_n, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{c\_n \cdot \left(c\_n \cdot \frac{1}{8}\right)} + \frac{-1}{8} \cdot c\_n, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \left(c\_n \cdot \frac{1}{8}\right) + \color{blue}{c\_n \cdot \frac{-1}{8}}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

    10. distribute-lft-out N/A

        \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{c\_n \cdot \left(c\_n \cdot \frac{1}{8} + \frac{-1}{8}\right)}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

    11. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{c\_n \cdot \left(c\_n \cdot \frac{1}{8} + \frac{-1}{8}\right)}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

    12. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \color{blue}{\mathsf{fma}\left(c\_n, \frac{1}{8}, \frac{-1}{8}\right)}, \frac{-1}{2} \cdot c\_n\right), 1\right) \]

    13. lower-*.f64 94.2

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

11. Applied rewrites 94.2%

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

12. Final simplification 94.2%

    \[\leadsto \mathsf{fma}\left(s, \mathsf{fma}\left(s, c\_n \cdot \mathsf{fma}\left(c\_n, 0.125, -0.125\right), c\_n \cdot -0.5\right), 1\right) \]
13. Add Preprocessing

Alternative 8: 94.1% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in c_p around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-+.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-pow.f64 N/A

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

5. Applied rewrites 94.2%

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

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}}} \]

   2. lower-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_n}}}{{\frac{1}{2}}^{c\_n}} \]

   3. sub-neg N/A

      \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)\right)}}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   5. neg-mul-1 N/A

      \[\leadsto \frac{{\left(1 + \left(\mathsf{neg}\left(\frac{1}{1 + e^{\color{blue}{-1 \cdot s}}}\right)\right)\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   6. distribute-neg-frac N/A

      \[\leadsto \frac{{\left(1 + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{1 + e^{-1 \cdot s}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. neg-mul-1 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   12. lower-neg.f64 N/A

       \[\leadsto \frac{{\left(1 + \frac{-1}{1 + e^{\color{blue}{\mathsf{neg}\left(s\right)}}}\right)}^{c\_n}}{{\frac{1}{2}}^{c\_n}} \]

   13. lower-pow.f64 91.8

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

8. Applied rewrites 91.8%

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

9. Taylor expanded in s around 0

   \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot \left(c\_n \cdot s\right)} \]
10. Step-by-step derivation

    1. associate-*r* N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(s, \frac{-1}{2} \cdot c\_n, 1\right)} \]

    5. lower-*.f64 94.2

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

11. Applied rewrites 94.2%

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

12. Final simplification 94.2%

    \[\leadsto \mathsf{fma}\left(s, c\_n \cdot -0.5, 1\right) \]
13. Add Preprocessing

Alternative 9: 94.1% accurate, 896.0× 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 91.0%

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

3. Taylor expanded in c_n around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-neg.f64 93.4

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

5. Applied rewrites 93.4%

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

6. Taylor expanded in c_p around 0

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

8. Applied rewrites 94.2%

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

Developer Target 1: 96.5% accurate, 1.4× 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 2024219 
(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))))