Harley's example

Percentage Accurate: 90.7% → 96.3%

Time: 52.0s

Alternatives: 7

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 6.033172460982282e+53
  2. c_n = 1.3402758776115743e-159
  3. t = 8.478884435320671e-306
  4. s = -7.064889964116292e-107

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(e^{-t}\right)\\ t_2 := e^{-s}\\ \mathbf{if}\;c\_p \leq 5.4 \cdot 10^{-188}:\\ \;\;\;\;\frac{{\left(1 - {\left(t\_2 + 1\right)}^{-1}\right)}^{c\_n}}{{\left(e^{-t\_1} + 1\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(t\_2\right)\right) + t\_1, c\_n \cdot \mathsf{fma}\left(0.5, t, -0.5 \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (log1p (exp (- t)))) (t_2 (exp (- s))))
   (if (<= c_p 5.4e-188)
     (/
      (pow (- 1.0 (pow (+ t_2 1.0) -1.0)) c_n)
      (pow (+ (exp (- t_1)) 1.0) c_n))
     (exp (fma c_p (+ (- (log1p t_2)) t_1) (* c_n (fma 0.5 t (* -0.5 s))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = log1p(exp(-t));
	double t_2 = exp(-s);
	double tmp;
	if (c_p <= 5.4e-188) {
		tmp = pow((1.0 - pow((t_2 + 1.0), -1.0)), c_n) / pow((exp(-t_1) + 1.0), c_n);
	} else {
		tmp = exp(fma(c_p, (-log1p(t_2) + t_1), (c_n * fma(0.5, t, (-0.5 * s)))));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = log1p(exp(Float64(-t)))
	t_2 = exp(Float64(-s))
	tmp = 0.0
	if (c_p <= 5.4e-188)
		tmp = Float64((Float64(1.0 - (Float64(t_2 + 1.0) ^ -1.0)) ^ c_n) / (Float64(exp(Float64(-t_1)) + 1.0) ^ c_n));
	else
		tmp = exp(fma(c_p, Float64(Float64(-log1p(t_2)) + t_1), Float64(c_n * fma(0.5, t, Float64(-0.5 * s)))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-s)], $MachinePrecision]}, If[LessEqual[c$95$p, 5.4e-188], N[(N[Power[N[(1.0 - N[Power[N[(t$95$2 + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[N[(N[Exp[(-t$95$1)], $MachinePrecision] + 1.0), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[Exp[N[(c$95$p * N[((-N[Log[1 + t$95$2], $MachinePrecision]) + t$95$1), $MachinePrecision] + N[(c$95$n * N[(0.5 * t + N[(-0.5 * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 5.4000000000000002e-188

   1. Initial program 89.1%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-pow.f64 N/A

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 98.7%

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

   if 5.4000000000000002e-188 < c_p

   1. Initial program 89.1%

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in t around 0

      \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\left(\left(\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + \frac{1}{2} \cdot t\right) - \log \frac{1}{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot t + \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)} - \log \frac{1}{2}\right)\right)} \]

      2. associate--l+ N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\left(\frac{1}{2} \cdot t + \left(\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}\right)\right)}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, t, \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}\right)}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \color{blue}{\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}}\right)\right)} \]

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in s around 0

      \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \frac{-1}{2} \cdot s\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 5.4 \cdot 10^{-188}:\\ \;\;\;\;\frac{{\left(1 - {\left(e^{-s} + 1\right)}^{-1}\right)}^{c\_n}}{{\left(e^{-\mathsf{log1p}\left(e^{-t}\right)} + 1\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) + \mathsf{log1p}\left(e^{-t}\right), c\_n \cdot \mathsf{fma}\left(0.5, t, -0.5 \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c\_n \cdot \mathsf{fma}\left(0.5, t, -0.5 \cdot s\right)\\ t_2 := \mathsf{log1p}\left(e^{-t}\right)\\ \mathbf{if}\;c\_p \leq 2 \cdot 10^{-43}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, t\_2 - \log 2, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) + t\_2, t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (* c_n (fma 0.5 t (* -0.5 s)))) (t_2 (log1p (exp (- t)))))
   (if (<= c_p 2e-43)
     (exp (fma c_p (- t_2 (log 2.0)) t_1))
     (exp (fma c_p (+ (- (log1p (exp (- s)))) t_2) t_1)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = c_n * fma(0.5, t, (-0.5 * s));
	double t_2 = log1p(exp(-t));
	double tmp;
	if (c_p <= 2e-43) {
		tmp = exp(fma(c_p, (t_2 - log(2.0)), t_1));
	} else {
		tmp = exp(fma(c_p, (-log1p(exp(-s)) + t_2), t_1));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(c_n * fma(0.5, t, Float64(-0.5 * s)))
	t_2 = log1p(exp(Float64(-t)))
	tmp = 0.0
	if (c_p <= 2e-43)
		tmp = exp(fma(c_p, Float64(t_2 - log(2.0)), t_1));
	else
		tmp = exp(fma(c_p, Float64(Float64(-log1p(exp(Float64(-s)))) + t_2), t_1));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(c$95$n * N[(0.5 * t + N[(-0.5 * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$p, 2e-43], N[Exp[N[(c$95$p * N[(t$95$2 - N[Log[2.0], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision], N[Exp[N[(c$95$p * N[((-N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]) + t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 2.00000000000000015e-43

   1. Initial program 91.4%

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in t around 0

      \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\left(\left(\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + \frac{1}{2} \cdot t\right) - \log \frac{1}{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot t + \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)} - \log \frac{1}{2}\right)\right)} \]

      2. associate--l+ N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\left(\frac{1}{2} \cdot t + \left(\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}\right)\right)}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, t, \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}\right)}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \color{blue}{\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}}\right)\right)} \]

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in s around 0

      \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \frac{-1}{2} \cdot s\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.8%

       \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \mathsf{fma}\left(0.5, t, -0.5 \cdot s\right)\right)} \]

   11. Taylor expanded in s around 0

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

       1. lower--.f64 N/A

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

       2. lower-log1p.f64 N/A

          \[\leadsto e^{\mathsf{fma}\left(c\_p, \color{blue}{\mathsf{log1p}\left(e^{\mathsf{neg}\left(t\right)}\right)} - \log 2, c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \frac{-1}{2} \cdot s\right)\right)} \]

       3. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(\color{blue}{e^{\mathsf{neg}\left(t\right)}}\right) - \log 2, c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \frac{-1}{2} \cdot s\right)\right)} \]

       4. lower-neg.f64 N/A

          \[\leadsto e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{\color{blue}{-t}}\right) - \log 2, c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \frac{-1}{2} \cdot s\right)\right)} \]

       5. lower-log.f64 97.9

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

   13. Applied rewrites 97.9%

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

   if 2.00000000000000015e-43 < c_p

   1. Initial program 79.6%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

      \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\left(\left(\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + \frac{1}{2} \cdot t\right) - \log \frac{1}{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot t + \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)} - \log \frac{1}{2}\right)\right)} \]

      2. associate--l+ N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\left(\frac{1}{2} \cdot t + \left(\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}\right)\right)}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, t, \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}\right)}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \color{blue}{\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}}\right)\right)} \]

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in s around 0

      \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \frac{-1}{2} \cdot s\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-43}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \log 2, c\_n \cdot \mathsf{fma}\left(0.5, t, -0.5 \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) + \mathsf{log1p}\left(e^{-t}\right), c\_n \cdot \mathsf{fma}\left(0.5, t, -0.5 \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_n \leq 6 \cdot 10^{-11}:\\ \;\;\;\;\frac{{\left(1 - {\left(e^{-s} + 1\right)}^{-1}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{{\left({\left(e^{-t} + 1\right)}^{-1}\right)}^{c\_p}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_n 6e-11)
   (/ (pow (- 1.0 (pow (+ (exp (- s)) 1.0) -1.0)) c_n) (pow 0.5 c_n))
   (/
    (pow (fma (fma (fma -0.16666666666666666 s 0.5) s -1.0) s 2.0) (- c_p))
    (pow (pow (+ (exp (- t)) 1.0) -1.0) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_n <= 6e-11) {
		tmp = pow((1.0 - pow((exp(-s) + 1.0), -1.0)), c_n) / pow(0.5, c_n);
	} else {
		tmp = pow(fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0), -c_p) / pow(pow((exp(-t) + 1.0), -1.0), c_p);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_n <= 6e-11)
		tmp = Float64((Float64(1.0 - (Float64(exp(Float64(-s)) + 1.0) ^ -1.0)) ^ c_n) / (0.5 ^ c_n));
	else
		tmp = Float64((fma(fma(fma(-0.16666666666666666, s, 0.5), s, -1.0), s, 2.0) ^ Float64(-c_p)) / ((Float64(exp(Float64(-t)) + 1.0) ^ -1.0) ^ c_p));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$n, 6e-11], N[(N[Power[N[(1.0 - N[Power[N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[0.5, c$95$n], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(-0.16666666666666666 * s + 0.5), $MachinePrecision] * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[N[Power[N[(N[Exp[(-t)], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_n < 6e-11

   1. Initial program 94.5%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-pow.f64 N/A

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 97.7%

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

   if 6e-11 < c_n

   1. Initial program 34.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2 + s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.6%

      \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)}\right)}^{c\_p}}{{\left(\frac{1}{e^{-t} + 1}\right)}^{c\_p}} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_n \leq 6 \cdot 10^{-11}:\\ \;\;\;\;\frac{{\left(1 - {\left(e^{-s} + 1\right)}^{-1}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right), s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{{\left({\left(e^{-t} + 1\right)}^{-1}\right)}^{c\_p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{{\left({\left(e^{-t} + 1\right)}^{-1}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, -\mathsf{log1p}\left(e^{-s}\right), \left(-c\_p\right) \cdot \log 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 7e-6)
   (/
    (pow (fma (fma 0.5 s -1.0) s 2.0) (- c_p))
    (pow (pow (+ (exp (- t)) 1.0) -1.0) c_p))
   (exp (fma c_p (- (log1p (exp (- s)))) (* (- c_p) (log 0.5))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 7e-6) {
		tmp = pow(fma(fma(0.5, s, -1.0), s, 2.0), -c_p) / pow(pow((exp(-t) + 1.0), -1.0), c_p);
	} else {
		tmp = exp(fma(c_p, -log1p(exp(-s)), (-c_p * log(0.5))));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 7e-6)
		tmp = Float64((fma(fma(0.5, s, -1.0), s, 2.0) ^ Float64(-c_p)) / ((Float64(exp(Float64(-t)) + 1.0) ^ -1.0) ^ c_p));
	else
		tmp = exp(fma(c_p, Float64(-log1p(exp(Float64(-s)))), Float64(Float64(-c_p) * log(0.5))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 7e-6], N[(N[Power[N[(N[(0.5 * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[N[Power[N[(N[Exp[(-t)], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision], N[Exp[N[(c$95$p * (-N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]) + N[((-c$95$p) * N[Log[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 6.99999999999999989e-6

   1. Initial program 92.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 96.9%

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

   10. Applied rewrites 96.9%

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

   if 6.99999999999999989e-6 < c_p

   1. Initial program 52.6%

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

   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. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 52.6

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.6%

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

   10. Applied rewrites 92.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(-c\_p\right)}}{{\left({\left(e^{-t} + 1\right)}^{-1}\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, -\mathsf{log1p}\left(e^{-s}\right), \left(-c\_p\right) \cdot \log 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -750000000:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, -\mathsf{log1p}\left(e^{-s}\right), \left(-c\_p\right) \cdot \log 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \log 2, c\_n \cdot \mathsf{fma}\left(0.5, t, -0.5 \cdot s\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -750000000.0)
   (exp (fma c_p (- (log1p (exp (- s)))) (* (- c_p) (log 0.5))))
   (exp
    (fma
     c_p
     (- (log1p (exp (- t))) (log 2.0))
     (* c_n (fma 0.5 t (* -0.5 s)))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -750000000.0) {
		tmp = exp(fma(c_p, -log1p(exp(-s)), (-c_p * log(0.5))));
	} else {
		tmp = exp(fma(c_p, (log1p(exp(-t)) - log(2.0)), (c_n * fma(0.5, t, (-0.5 * s)))));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -750000000.0)
		tmp = exp(fma(c_p, Float64(-log1p(exp(Float64(-s)))), Float64(Float64(-c_p) * log(0.5))));
	else
		tmp = exp(fma(c_p, Float64(log1p(exp(Float64(-t))) - log(2.0)), Float64(c_n * fma(0.5, t, Float64(-0.5 * s)))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -750000000.0], N[Exp[N[(c$95$p * (-N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]) + N[((-c$95$p) * N[Log[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(c$95$p * N[(N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision] - N[Log[2.0], $MachinePrecision]), $MachinePrecision] + N[(c$95$n * N[(0.5 * t + N[(-0.5 * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -7.5e8

   1. Initial program 60.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 60.0

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 60.0%

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

   10. Applied rewrites 100.0%

       \[\leadsto e^{\mathsf{fma}\left(c\_p, -\mathsf{log1p}\left(e^{-s}\right), -c\_p \cdot \log 0.5\right)} \]

   if -7.5e8 < s

   1. Initial program 89.7%

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in t around 0

      \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\left(\left(\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) + \frac{1}{2} \cdot t\right) - \log \frac{1}{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot t + \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)\right)} - \log \frac{1}{2}\right)\right)} \]

      2. associate--l+ N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\left(\frac{1}{2} \cdot t + \left(\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}\right)\right)}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, t, \log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}\right)}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \color{blue}{\log \left(1 - \frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right) - \log \frac{1}{2}}\right)\right)} \]

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in s around 0

      \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \frac{-1}{2} \cdot s\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.5%

       \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) - \left(-\mathsf{log1p}\left(e^{-t}\right)\right), c\_n \cdot \mathsf{fma}\left(0.5, t, -0.5 \cdot s\right)\right)} \]

   11. Taylor expanded in s around 0

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

       1. lower--.f64 N/A

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

       2. lower-log1p.f64 N/A

          \[\leadsto e^{\mathsf{fma}\left(c\_p, \color{blue}{\mathsf{log1p}\left(e^{\mathsf{neg}\left(t\right)}\right)} - \log 2, c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \frac{-1}{2} \cdot s\right)\right)} \]

       3. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(\color{blue}{e^{\mathsf{neg}\left(t\right)}}\right) - \log 2, c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \frac{-1}{2} \cdot s\right)\right)} \]

       4. lower-neg.f64 N/A

          \[\leadsto e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{\color{blue}{-t}}\right) - \log 2, c\_n \cdot \mathsf{fma}\left(\frac{1}{2}, t, \frac{-1}{2} \cdot s\right)\right)} \]

       5. lower-log.f64 97.8

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

   13. Applied rewrites 97.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq -750000000:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, -\mathsf{log1p}\left(e^{-s}\right), \left(-c\_p\right) \cdot \log 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{-t}\right) - \log 2, c\_n \cdot \mathsf{fma}\left(0.5, t, -0.5 \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.9% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.1%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-pow.f64 N/A

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

5. Applied rewrites 93.7%

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

6. Taylor expanded in s around 0

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

8. Applied rewrites 93.6%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 93.7%

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

Alternative 7: 93.9% 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 89.1%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 90.6

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

5. Applied rewrites 90.6%

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

6. Taylor expanded in c_p around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 93.6%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 96.2% 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 2024298 
(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))))