Harley's example

Percentage Accurate: 90.5% → 99.0%

Time: 2.3min

Alternatives: 7

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 2.00000000000000007e154
  2. c_n = 2
  3. t = 0.0
  4. s = 0.0

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.5% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 5 \cdot 10^{-30}:\\ \;\;\;\;{\left(\sqrt{e^{\left(\left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 5e-30)
   (pow
    (sqrt
     (exp
      (+
       (-
        (- (* c_n (log1p (/ 1.0 (+ (exp s) 1.0)))) (* c_p (log1p (exp s))))
        (* c_n (log1p (/ 1.0 (+ 1.0 (exp t))))))
       (* c_p (log1p (exp t))))))
    2.0)
   (exp (* c_p (- (log1p (exp (- t))) (log1p (exp (- s))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 5e-30) {
		tmp = pow(sqrt(exp(((((c_n * log1p((1.0 / (exp(s) + 1.0)))) - (c_p * log1p(exp(s)))) - (c_n * log1p((1.0 / (1.0 + exp(t)))))) + (c_p * log1p(exp(t)))))), 2.0);
	} else {
		tmp = exp((c_p * (log1p(exp(-t)) - log1p(exp(-s)))));
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 5e-30) {
		tmp = Math.pow(Math.sqrt(Math.exp(((((c_n * Math.log1p((1.0 / (Math.exp(s) + 1.0)))) - (c_p * Math.log1p(Math.exp(s)))) - (c_n * Math.log1p((1.0 / (1.0 + Math.exp(t)))))) + (c_p * Math.log1p(Math.exp(t)))))), 2.0);
	} else {
		tmp = Math.exp((c_p * (Math.log1p(Math.exp(-t)) - Math.log1p(Math.exp(-s)))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 5e-30:
		tmp = math.pow(math.sqrt(math.exp(((((c_n * math.log1p((1.0 / (math.exp(s) + 1.0)))) - (c_p * math.log1p(math.exp(s)))) - (c_n * math.log1p((1.0 / (1.0 + math.exp(t)))))) + (c_p * math.log1p(math.exp(t)))))), 2.0)
	else:
		tmp = math.exp((c_p * (math.log1p(math.exp(-t)) - math.log1p(math.exp(-s)))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 5e-30)
		tmp = sqrt(exp(Float64(Float64(Float64(Float64(c_n * log1p(Float64(1.0 / Float64(exp(s) + 1.0)))) - Float64(c_p * log1p(exp(s)))) - Float64(c_n * log1p(Float64(1.0 / Float64(1.0 + exp(t)))))) + Float64(c_p * log1p(exp(t)))))) ^ 2.0;
	else
		tmp = exp(Float64(c_p * Float64(log1p(exp(Float64(-t))) - log1p(exp(Float64(-s))))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 5e-30], N[Power[N[Sqrt[N[Exp[N[(N[(N[(N[(c$95$n * N[Log[1 + N[(1.0 / N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(c$95$p * N[Log[1 + N[Exp[s], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c$95$n * N[Log[1 + N[(1.0 / N[(1.0 + N[Exp[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * N[Log[1 + N[Exp[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], N[Exp[N[(c$95$p * N[(N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 4.99999999999999972e-30

   1. Initial program 90.9%

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

      1. associate-/l/ 90.9%

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

   3. Simplified 90.9%

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

   6. Applied egg-rr 99.2%

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

   if 4.99999999999999972e-30 < c_p

   1. Initial program 57.9%

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

      1. associate-/l/ 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in c_n around 0 65.7%

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

      1. add-exp-log 65.7%

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

      2. log-div 65.7%

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

      3. log-pow 69.0%

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

      4. log-rec 69.0%

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

      5. log1p-define 69.0%

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

      6. log-pow 99.4%

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

      7. neg-log 99.4%

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

      8. log1p-define 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. log1p-undefine 99.4%

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

      2. log-rec 99.4%

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

      3. distribute-lft-out-- 99.4%

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

      4. log-rec 99.4%

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

      5. log1p-undefine 99.4%

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

   9. Simplified 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 5 \cdot 10^{-30}:\\ \;\;\;\;{\left(\sqrt{e^{\left(\left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 3 \cdot 10^{-29}:\\ \;\;\;\;{\left(\frac{2}{e^{s} + 1}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 3e-29)
   (pow (/ 2.0 (+ (exp s) 1.0)) c_p)
   (exp (* c_p (- (log1p (exp (- t))) (log1p (exp (- s))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 3e-29) {
		tmp = pow((2.0 / (exp(s) + 1.0)), c_p);
	} else {
		tmp = exp((c_p * (log1p(exp(-t)) - log1p(exp(-s)))));
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 3e-29) {
		tmp = Math.pow((2.0 / (Math.exp(s) + 1.0)), c_p);
	} else {
		tmp = Math.exp((c_p * (Math.log1p(Math.exp(-t)) - Math.log1p(Math.exp(-s)))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 3e-29:
		tmp = math.pow((2.0 / (math.exp(s) + 1.0)), c_p)
	else:
		tmp = math.exp((c_p * (math.log1p(math.exp(-t)) - math.log1p(math.exp(-s)))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 3e-29)
		tmp = Float64(2.0 / Float64(exp(s) + 1.0)) ^ c_p;
	else
		tmp = exp(Float64(c_p * Float64(log1p(exp(Float64(-t))) - log1p(exp(Float64(-s))))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 3e-29], N[Power[N[(2.0 / N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[Exp[N[(c$95$p * N[(N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 3.0000000000000003e-29

   1. Initial program 90.9%

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

      1. associate-/l/ 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in c_n around 0 93.8%

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

      1. add-exp-log 93.8%

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

      2. log-div 93.8%

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

      3. log-pow 93.8%

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

      4. log-rec 93.8%

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

      5. log1p-define 93.8%

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

      6. log-pow 93.8%

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

      7. neg-log 93.8%

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

      8. log1p-define 93.8%

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

   7. Applied egg-rr 93.8%

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

      1. log1p-undefine 93.8%

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

      2. log-rec 93.8%

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

      3. distribute-lft-out-- 93.8%

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

      4. log-rec 93.8%

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

      5. log1p-undefine 93.8%

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

   9. Simplified 93.8%

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

   10. Taylor expanded in t around 0 94.7%

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

       1. log1p-define 94.7%

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

   12. Simplified 94.7%

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

       1. *-commutative 94.7%

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

       2. exp-prod 94.7%

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

       3. exp-diff 94.7%

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

       4. log1p-undefine 94.7%

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

       5. add-exp-log 94.7%

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

       6. rem-exp-log 94.7%

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

       7. +-commutative 94.7%

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

       8. add-sqr-sqrt 47.9%

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

       9. sqrt-unprod 97.4%

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

       10. sqr-neg 97.4%

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

       11. sqrt-unprod 49.5%

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

       12. add-sqr-sqrt 98.3%

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

   14. Applied egg-rr 98.3%

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

   if 3.0000000000000003e-29 < c_p

   1. Initial program 57.9%

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

      1. associate-/l/ 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in c_n around 0 65.7%

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

      1. add-exp-log 65.7%

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

      2. log-div 65.7%

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

      3. log-pow 69.0%

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

      4. log-rec 69.0%

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

      5. log1p-define 69.0%

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

      6. log-pow 99.4%

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

      7. neg-log 99.4%

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

      8. log1p-define 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. log1p-undefine 99.4%

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

      2. log-rec 99.4%

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

      3. distribute-lft-out-- 99.4%

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

      4. log-rec 99.4%

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

      5. log1p-undefine 99.4%

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

   9. Simplified 99.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 3 \cdot 10^{-29}:\\ \;\;\;\;{\left(\frac{2}{e^{s} + 1}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\mathsf{log1p}\left(e^{-t}\right) - \mathsf{log1p}\left(e^{-s}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-29}:\\ \;\;\;\;{\left(\frac{2}{e^{s} + 1}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(\log 2 - \mathsf{log1p}\left(e^{-s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 2e-29)
   (pow (/ 2.0 (+ (exp s) 1.0)) c_p)
   (exp (* c_p (- (log 2.0) (log1p (exp (- s))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 2e-29) {
		tmp = pow((2.0 / (exp(s) + 1.0)), c_p);
	} else {
		tmp = exp((c_p * (log(2.0) - log1p(exp(-s)))));
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 2e-29) {
		tmp = Math.pow((2.0 / (Math.exp(s) + 1.0)), c_p);
	} else {
		tmp = Math.exp((c_p * (Math.log(2.0) - Math.log1p(Math.exp(-s)))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 2e-29:
		tmp = math.pow((2.0 / (math.exp(s) + 1.0)), c_p)
	else:
		tmp = math.exp((c_p * (math.log(2.0) - math.log1p(math.exp(-s)))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 2e-29)
		tmp = Float64(2.0 / Float64(exp(s) + 1.0)) ^ c_p;
	else
		tmp = exp(Float64(c_p * Float64(log(2.0) - log1p(exp(Float64(-s))))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 2e-29], N[Power[N[(2.0 / N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[Exp[N[(c$95$p * N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1.99999999999999989e-29

   1. Initial program 90.9%

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

      1. associate-/l/ 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in c_n around 0 93.8%

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

      1. add-exp-log 93.8%

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

      2. log-div 93.8%

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

      3. log-pow 93.8%

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

      4. log-rec 93.8%

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

      5. log1p-define 93.8%

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

      6. log-pow 93.8%

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

      7. neg-log 93.8%

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

      8. log1p-define 93.8%

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

   7. Applied egg-rr 93.8%

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

      1. log1p-undefine 93.8%

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

      2. log-rec 93.8%

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

      3. distribute-lft-out-- 93.8%

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

      4. log-rec 93.8%

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

      5. log1p-undefine 93.8%

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

   9. Simplified 93.8%

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

   10. Taylor expanded in t around 0 94.7%

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

       1. log1p-define 94.7%

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

   12. Simplified 94.7%

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

       1. *-commutative 94.7%

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

       2. exp-prod 94.7%

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

       3. exp-diff 94.7%

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

       4. log1p-undefine 94.7%

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

       5. add-exp-log 94.7%

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

       6. rem-exp-log 94.7%

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

       7. +-commutative 94.7%

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

       8. add-sqr-sqrt 47.9%

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

       9. sqrt-unprod 97.4%

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

       10. sqr-neg 97.4%

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

       11. sqrt-unprod 49.5%

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

       12. add-sqr-sqrt 98.3%

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

   14. Applied egg-rr 98.3%

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

   if 1.99999999999999989e-29 < c_p

   1. Initial program 57.9%

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

      1. associate-/l/ 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in c_n around 0 65.7%

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

      1. add-exp-log 65.7%

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

      2. log-div 65.7%

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

      3. log-pow 69.0%

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

      4. log-rec 69.0%

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

      5. log1p-define 69.0%

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

      6. log-pow 99.4%

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

      7. neg-log 99.4%

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

      8. log1p-define 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. log1p-undefine 99.4%

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

      2. log-rec 99.4%

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

      3. distribute-lft-out-- 99.4%

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

      4. log-rec 99.4%

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

      5. log1p-undefine 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in t around 0 96.8%

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

       1. log1p-define 96.8%

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

   12. Simplified 96.8%

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

Alternative 4: 97.8% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate-/l/ 86.0%

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

3. Simplified 86.0%

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

5. Taylor expanded in c_n around 0 89.7%

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

   1. add-exp-log 89.7%

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

   2. log-div 89.7%

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

   3. log-pow 90.2%

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

   4. log-rec 90.2%

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

   5. log1p-define 90.2%

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

   6. log-pow 94.7%

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

   7. neg-log 94.7%

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

   8. log1p-define 94.7%

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

7. Applied egg-rr 94.7%

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

   1. log1p-undefine 94.7%

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

   2. log-rec 94.7%

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

   3. distribute-lft-out-- 94.7%

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

   4. log-rec 94.7%

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

   5. log1p-undefine 94.7%

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

9. Simplified 94.7%

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

10. Taylor expanded in t around 0 95.1%

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

    1. log1p-define 95.1%

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

12. Simplified 95.1%

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

13. Taylor expanded in s around 0 96.4%

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

    1. *-commutative 96.4%

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

15. Simplified 96.4%

    \[\leadsto e^{c\_p \cdot \color{blue}{\left(s \cdot \left(0.5 + s \cdot -0.125\right)\right)}} \]
16. Add Preprocessing

Alternative 5: 96.4% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate-/l/ 86.0%

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

3. Simplified 86.0%

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

5. Taylor expanded in c_n around 0 89.7%

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

   1. add-exp-log 89.7%

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

   2. log-div 89.7%

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

   3. log-pow 90.2%

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

   4. log-rec 90.2%

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

   5. log1p-define 90.2%

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

   6. log-pow 94.7%

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

   7. neg-log 94.7%

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

   8. log1p-define 94.7%

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

7. Applied egg-rr 94.7%

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

   1. log1p-undefine 94.7%

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

   2. log-rec 94.7%

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

   3. distribute-lft-out-- 94.7%

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

   4. log-rec 94.7%

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

   5. log1p-undefine 94.7%

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

9. Simplified 94.7%

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

10. Taylor expanded in t around 0 95.1%

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

    1. log1p-define 95.1%

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

12. Simplified 95.1%

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

13. Taylor expanded in s around 0 95.6%

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

    1. *-commutative 95.6%

       \[\leadsto e^{\color{blue}{\left(c\_p \cdot s\right) \cdot 0.5}} \]

    2. associate-*l* 95.6%

       \[\leadsto e^{\color{blue}{c\_p \cdot \left(s \cdot 0.5\right)}} \]

15. Simplified 95.6%

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

Alternative 6: 94.0% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate-/l/ 86.0%

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

3. Simplified 86.0%

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

5. Taylor expanded in c_n around 0 89.7%

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

6. Taylor expanded in s around 0 89.5%

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

7. Taylor expanded in t around 0 93.1%

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

Alternative 7: 94.0% accurate, 835.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

   1. associate-/l/ 86.0%

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

3. Simplified 86.0%

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

5. Taylor expanded in c_n around 0 89.7%

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

6. Taylor expanded in c_p around 0 93.0%

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

Developer Target 1: 96.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024165 
(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))))