Harley's example

Percentage Accurate: 91.6% → 98.9%

Time: 1.3min

Alternatives: 7

Speedup: 835.0×

• could not determine a ground truth

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

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-21}:\\ \;\;\;\;e^{\left(c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \mathsf{log1p}\left(e^{-t}\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 2e-21)
   (exp
    (+
     (-
      (*
       c_n
       (- (log1p (/ 1.0 (+ (exp s) 1.0))) (log1p (/ 1.0 (+ 1.0 (exp t))))))
      (* c_p (log1p (exp s))))
     (* c_p (log1p (exp t)))))
   (exp (- (* c_p (log1p (exp (- t)))) (* c_p (log1p (exp (- s))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 2e-21) {
		tmp = exp((((c_n * (log1p((1.0 / (exp(s) + 1.0))) - log1p((1.0 / (1.0 + exp(t)))))) - (c_p * log1p(exp(s)))) + (c_p * log1p(exp(t)))));
	} else {
		tmp = exp(((c_p * log1p(exp(-t))) - (c_p * 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-21) {
		tmp = Math.exp((((c_n * (Math.log1p((1.0 / (Math.exp(s) + 1.0))) - Math.log1p((1.0 / (1.0 + Math.exp(t)))))) - (c_p * Math.log1p(Math.exp(s)))) + (c_p * Math.log1p(Math.exp(t)))));
	} else {
		tmp = Math.exp(((c_p * Math.log1p(Math.exp(-t))) - (c_p * Math.log1p(Math.exp(-s)))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 2e-21:
		tmp = math.exp((((c_n * (math.log1p((1.0 / (math.exp(s) + 1.0))) - math.log1p((1.0 / (1.0 + math.exp(t)))))) - (c_p * math.log1p(math.exp(s)))) + (c_p * math.log1p(math.exp(t)))))
	else:
		tmp = math.exp(((c_p * math.log1p(math.exp(-t))) - (c_p * math.log1p(math.exp(-s)))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 2e-21)
		tmp = exp(Float64(Float64(Float64(c_n * Float64(log1p(Float64(1.0 / Float64(exp(s) + 1.0))) - log1p(Float64(1.0 / Float64(1.0 + exp(t)))))) - Float64(c_p * log1p(exp(s)))) + Float64(c_p * log1p(exp(t)))));
	else
		tmp = exp(Float64(Float64(c_p * log1p(exp(Float64(-t)))) - Float64(c_p * log1p(exp(Float64(-s))))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 2e-21], N[Exp[N[(N[(N[(c$95$n * N[(N[Log[1 + N[(1.0 / N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(1.0 / N[(1.0 + N[Exp[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c$95$p * N[Log[1 + N[Exp[s], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c$95$p * N[Log[1 + N[Exp[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(c$95$p * N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(c$95$p * N[Log[1 + N[Exp[(-s)], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1.99999999999999982e-21

   1. Initial program 92.3%

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

      1. associate-/l/ 92.3%

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

   3. Simplified 92.3%

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

   6. Applied egg-rr 99.0%

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

      1. *-lft-identity 99.0%

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

      2. associate--l+ 99.0%

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

      3. distribute-lft-out-- 99.0%

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

   8. Simplified 99.0%

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

   if 1.99999999999999982e-21 < c_p

   1. Initial program 53.8%

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

      1. associate-/l/ 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in c_n around 0 57.7%

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

      1. add-exp-log 57.7%

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

      2. log-div 57.7%

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

      3. log-pow 58.4%

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

      4. log-rec 58.4%

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

      5. log1p-define 58.4%

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

      6. log-pow 98.6%

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

      7. log-rec 98.6%

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

      8. log1p-define 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 2 \cdot 10^{-21}:\\ \;\;\;\;e^{\left(c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \mathsf{log1p}\left(e^{-t}\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s}\\ t_2 := e^{-t}\\ \mathbf{if}\;c\_n \leq 10^{-19}:\\ \;\;\;\;e^{c\_p \cdot \mathsf{log1p}\left(t\_2\right) - c\_p \cdot \mathsf{log1p}\left(t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_n \cdot \mathsf{log1p}\left(\frac{-1}{1 + t\_1}\right) - c\_n \cdot \mathsf{log1p}\left(\frac{-1}{1 + t\_2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- s))) (t_2 (exp (- t))))
   (if (<= c_n 1e-19)
     (exp (- (* c_p (log1p t_2)) (* c_p (log1p t_1))))
     (exp
      (-
       (* c_n (log1p (/ -1.0 (+ 1.0 t_1))))
       (* c_n (log1p (/ -1.0 (+ 1.0 t_2)))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-s);
	double t_2 = exp(-t);
	double tmp;
	if (c_n <= 1e-19) {
		tmp = exp(((c_p * log1p(t_2)) - (c_p * log1p(t_1))));
	} else {
		tmp = exp(((c_n * log1p((-1.0 / (1.0 + t_1)))) - (c_n * log1p((-1.0 / (1.0 + t_2))))));
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = Math.exp(-s);
	double t_2 = Math.exp(-t);
	double tmp;
	if (c_n <= 1e-19) {
		tmp = Math.exp(((c_p * Math.log1p(t_2)) - (c_p * Math.log1p(t_1))));
	} else {
		tmp = Math.exp(((c_n * Math.log1p((-1.0 / (1.0 + t_1)))) - (c_n * Math.log1p((-1.0 / (1.0 + t_2))))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = math.exp(-s)
	t_2 = math.exp(-t)
	tmp = 0
	if c_n <= 1e-19:
		tmp = math.exp(((c_p * math.log1p(t_2)) - (c_p * math.log1p(t_1))))
	else:
		tmp = math.exp(((c_n * math.log1p((-1.0 / (1.0 + t_1)))) - (c_n * math.log1p((-1.0 / (1.0 + t_2))))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-s))
	t_2 = exp(Float64(-t))
	tmp = 0.0
	if (c_n <= 1e-19)
		tmp = exp(Float64(Float64(c_p * log1p(t_2)) - Float64(c_p * log1p(t_1))));
	else
		tmp = exp(Float64(Float64(c_n * log1p(Float64(-1.0 / Float64(1.0 + t_1)))) - Float64(c_n * log1p(Float64(-1.0 / Float64(1.0 + t_2))))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-s)], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t)], $MachinePrecision]}, If[LessEqual[c$95$n, 1e-19], N[Exp[N[(N[(c$95$p * N[Log[1 + t$95$2], $MachinePrecision]), $MachinePrecision] - N[(c$95$p * N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(c$95$n * N[Log[1 + N[(-1.0 / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(c$95$n * N[Log[1 + N[(-1.0 / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c_n < 9.9999999999999998e-20

   1. Initial program 91.8%

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

      1. associate-/l/ 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in c_n around 0 94.4%

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

      1. add-exp-log 94.4%

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

      2. log-div 94.4%

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

      3. log-pow 94.4%

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

      4. log-rec 94.4%

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

      5. log1p-define 94.4%

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

      6. log-pow 98.6%

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

      7. log-rec 98.6%

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

      8. log1p-define 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 9.9999999999999998e-20 < c_n

   1. Initial program 59.2%

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

      1. associate-/l/ 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in c_p around 0 66.6%

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

      1. add-exp-log 66.6%

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

      2. log-div 66.6%

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

      3. log-pow 67.4%

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

      4. sub-neg 67.4%

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

      5. log1p-define 67.4%

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

      6. log-pow 99.6%

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

      7. sub-neg 99.6%

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

      8. log1p-define 99.6%

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

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{e^{c\_n \cdot \mathsf{log1p}\left(-\frac{1}{1 + e^{-s}}\right) - c\_n \cdot \mathsf{log1p}\left(-\frac{1}{1 + e^{-t}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_n \leq 10^{-19}:\\ \;\;\;\;e^{c\_p \cdot \mathsf{log1p}\left(e^{-t}\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_n \cdot \mathsf{log1p}\left(\frac{-1}{1 + e^{-s}}\right) - c\_n \cdot \mathsf{log1p}\left(\frac{-1}{1 + e^{-t}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ e^{c\_p \cdot \mathsf{log1p}\left(e^{-t}\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp (- (* c_p (log1p (exp (- t)))) (* c_p (log1p (exp (- s)))))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(((c_p * log1p(exp(-t))) - (c_p * log1p(exp(-s)))));
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp(((c_p * Math.log1p(Math.exp(-t))) - (c_p * Math.log1p(Math.exp(-s)))));
}

Python

def code(c_p, c_n, t, s):
	return math.exp(((c_p * math.log1p(math.exp(-t))) - (c_p * math.log1p(math.exp(-s)))))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(c_p * log1p(exp(Float64(-t)))) - Float64(c_p * log1p(exp(Float64(-s))))))
end

Wolfram

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

Derivation

1. Initial program 88.4%

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

   1. associate-/l/ 88.4%

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

3. Simplified 88.4%

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

5. Taylor expanded in c_n around 0 92.1%

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

   1. add-exp-log 92.1%

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

   2. log-div 92.1%

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

   3. log-pow 92.2%

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

   4. log-rec 92.2%

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

   5. log1p-define 92.2%

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

   6. log-pow 96.3%

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

   7. log-rec 96.3%

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

   8. log1p-define 96.3%

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

7. Applied egg-rr 96.3%

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

8. Final simplification 96.3%

   \[\leadsto e^{c\_p \cdot \mathsf{log1p}\left(e^{-t}\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)} \]
9. Add Preprocessing

Alternative 4: 96.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 4000000:\\ \;\;\;\;1 + t \cdot \left(c\_p \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 4000000.0)
   (+ 1.0 (* t (* c_p -0.5)))
   (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 4000000.0) {
		tmp = 1.0 + (t * (c_p * -0.5));
	} else {
		tmp = pow((1.0 / (1.0 + exp(-s))), c_p);
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (-s <= 4000000.0d0) then
        tmp = 1.0d0 + (t * (c_p * (-0.5d0)))
    else
        tmp = (1.0d0 / (1.0d0 + exp(-s))) ** c_p
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 4000000.0) {
		tmp = 1.0 + (t * (c_p * -0.5));
	} else {
		tmp = Math.pow((1.0 / (1.0 + Math.exp(-s))), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if -s <= 4000000.0:
		tmp = 1.0 + (t * (c_p * -0.5))
	else:
		tmp = math.pow((1.0 / (1.0 + math.exp(-s))), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 4000000.0)
		tmp = Float64(1.0 + Float64(t * Float64(c_p * -0.5)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-s)))) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (-s <= 4000000.0)
		tmp = 1.0 + (t * (c_p * -0.5));
	else
		tmp = (1.0 / (1.0 + exp(-s))) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 4000000.0], N[(1.0 + N[(t * N[(c$95$p * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 4e6

   1. Initial program 90.0%

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

      1. associate-/l/ 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in c_n around 0 93.9%

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

   6. Taylor expanded in c_p around 0 95.7%

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

      1. log-rec 95.7%

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

      2. log1p-undefine 95.7%

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

      3. log-rec 95.7%

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

      4. log1p-undefine 95.7%

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

   8. Simplified 95.7%

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

   9. Taylor expanded in t around 0 96.3%

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

   10. Taylor expanded in t around inf 97.1%

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

       1. associate-*r* 97.1%

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

       2. *-commutative 97.1%

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

   12. Simplified 97.1%

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

   if 4e6 < (neg.f64 s)

   1. Initial program 37.5%

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

      1. associate-/l/ 37.5%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in c_n around 0 37.5%

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

   6. Taylor expanded in c_p around 0 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 4000000:\\ \;\;\;\;1 + t \cdot \left(c\_p \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

   1. associate-/l/ 88.4%

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

3. Simplified 88.4%

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

5. Taylor expanded in c_n around 0 92.1%

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

6. Taylor expanded in c_p around 0 92.8%

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

   1. log-rec 92.8%

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

   2. log1p-undefine 92.8%

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

   3. log-rec 92.8%

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

   4. log1p-undefine 92.8%

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

8. Simplified 92.8%

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

9. Taylor expanded in t around 0 93.3%

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

10. Taylor expanded in t around inf 94.1%

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

    1. associate-*r* 94.1%

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

    2. *-commutative 94.1%

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

12. Simplified 94.1%

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

13. Final simplification 94.1%

    \[\leadsto 1 + t \cdot \left(c\_p \cdot -0.5\right) \]
14. Add Preprocessing

Alternative 6: 94.6% accurate, 119.3× speedup

Math

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

FPCore

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

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (0.5 * (c_n * 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_n * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

   1. associate-/l/ 88.4%

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

3. Simplified 88.4%

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

5. Taylor expanded in c_p around 0 91.0%

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

6. Taylor expanded in s around 0 92.1%

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

7. Taylor expanded in t around 0 94.1%

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

   1. *-commutative 94.1%

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

9. Simplified 94.1%

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

10. Final simplification 94.1%

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

Alternative 7: 94.6% 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 88.4%

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

   1. associate-/l/ 88.4%

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

3. Simplified 88.4%

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

5. Taylor expanded in c_n around 0 92.1%

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

6. Taylor expanded in c_p around 0 94.0%

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

Developer target: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024091 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

  :alt
  (* (pow (/ (+ 1.0 (exp (- t))) (+ 1.0 (exp (- s)))) c_p) (pow (/ (+ 1.0 (exp t)) (+ 1.0 (exp s))) c_n))

  (/ (* (pow (/ 1.0 (+ 1.0 (exp (- s)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- s))))) c_n)) (* (pow (/ 1.0 (+ 1.0 (exp (- t)))) c_p) (pow (- 1.0 (/ 1.0 (+ 1.0 (exp (- t))))) c_n))))