Harley's example

Percentage Accurate: 91.3% → 96.5%

Time: 2.4min

Alternatives: 5

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: 91.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ e^{\left(c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{e^{s} + 1}\right) - \mathsf{log1p}\left(\frac{1}{1 + e^{t}}\right)\right) - c\_p \cdot \mathsf{log1p}\left(e^{s}\right)\right) + c\_p \cdot \mathsf{log1p}\left(e^{t}\right)} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-/l/ 91.5%

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

3. Simplified 91.5%

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

6. Applied egg-rr 99.1%

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

   1. *-lft-identity 99.1%

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

   2. associate--l+ 99.1%

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

   3. distribute-lft-out-- 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 2: 97.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{s} + 1\\ \mathbf{if}\;c\_p \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;{t\_1}^{\left(-c\_p\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + \frac{-1}{t\_1}\right)}^{c\_n} \cdot {0.5}^{c\_n}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (+ (exp s) 1.0)))
   (if (<= c_p 6.6e-9)
     (pow t_1 (- c_p))
     (* (pow (+ 1.0 (/ -1.0 t_1)) c_n) (pow 0.5 c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(s) + 1.0;
	double tmp;
	if (c_p <= 6.6e-9) {
		tmp = pow(t_1, -c_p);
	} else {
		tmp = pow((1.0 + (-1.0 / t_1)), c_n) * pow(0.5, c_n);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = exp(s) + 1.0d0
    if (c_p <= 6.6d-9) then
        tmp = t_1 ** -c_p
    else
        tmp = ((1.0d0 + ((-1.0d0) / t_1)) ** c_n) * (0.5d0 ** c_n)
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = Math.exp(s) + 1.0;
	double tmp;
	if (c_p <= 6.6e-9) {
		tmp = Math.pow(t_1, -c_p);
	} else {
		tmp = Math.pow((1.0 + (-1.0 / t_1)), c_n) * Math.pow(0.5, c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = math.exp(s) + 1.0
	tmp = 0
	if c_p <= 6.6e-9:
		tmp = math.pow(t_1, -c_p)
	else:
		tmp = math.pow((1.0 + (-1.0 / t_1)), c_n) * math.pow(0.5, c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(exp(s) + 1.0)
	tmp = 0.0
	if (c_p <= 6.6e-9)
		tmp = t_1 ^ Float64(-c_p);
	else
		tmp = Float64((Float64(1.0 + Float64(-1.0 / t_1)) ^ c_n) * (0.5 ^ c_n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	t_1 = exp(s) + 1.0;
	tmp = 0.0;
	if (c_p <= 6.6e-9)
		tmp = t_1 ^ -c_p;
	else
		tmp = ((1.0 + (-1.0 / t_1)) ^ c_n) * (0.5 ^ c_n);
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[c$95$p, 6.6e-9], N[Power[t$95$1, (-c$95$p)], $MachinePrecision], N[(N[Power[N[(1.0 + N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] * N[Power[0.5, c$95$n], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 6.60000000000000037e-9

   1. Initial program 93.1%

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

      1. associate-/l/ 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in c_n around 0 95.2%

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

      1. *-un-lft-identity 95.2%

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

      2. inv-pow 95.2%

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

      3. pow-pow 95.2%

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

      4. add-sqr-sqrt 45.9%

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

      5. sqrt-unprod 97.6%

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

      6. sqr-neg 97.6%

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

      7. sqrt-unprod 51.6%

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

      8. add-sqr-sqrt 98.0%

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

   7. Applied egg-rr 98.0%

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

      1. *-lft-identity 98.0%

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

      2. neg-mul-1 98.0%

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

   9. Simplified 98.0%

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

   10. Taylor expanded in c_p around 0 98.8%

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

   if 6.60000000000000037e-9 < c_p

   1. Initial program 58.3%

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

      1. associate-/l/ 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in t around 0 58.3%

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

   6. Taylor expanded in c_p around 0 91.9%

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

      1. div-inv 91.9%

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

      2. add-sqr-sqrt 66.9%

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

      3. sqrt-unprod 91.9%

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

      4. sqr-neg 91.9%

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

      5. sqrt-unprod 25.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. pow-flip 100.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 100.0%

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

      10. sqr-neg 100.0%

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

      11. sqrt-unprod 100.0%

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

      12. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 6.6 \cdot 10^{-9}:\\ \;\;\;\;{\left(e^{s} + 1\right)}^{\left(-c\_p\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + \frac{-1}{e^{s} + 1}\right)}^{c\_n} \cdot {0.5}^{c\_n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_n <= 2.0) {
		tmp = pow((1.0 + (1.0 / (-1.0 - (1.0 + (s * ((s * 0.5) + -1.0)))))), c_n) / pow(0.5, c_n);
	} else {
		tmp = pow((exp(s) + 1.0), -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 (c_n <= 2.0d0) then
        tmp = ((1.0d0 + (1.0d0 / ((-1.0d0) - (1.0d0 + (s * ((s * 0.5d0) + (-1.0d0))))))) ** c_n) / (0.5d0 ** c_n)
    else
        tmp = (exp(s) + 1.0d0) ** -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 (c_n <= 2.0) {
		tmp = Math.pow((1.0 + (1.0 / (-1.0 - (1.0 + (s * ((s * 0.5) + -1.0)))))), c_n) / Math.pow(0.5, c_n);
	} else {
		tmp = Math.pow((Math.exp(s) + 1.0), -c_p);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_n <= 2.0)
		tmp = ((1.0 + (1.0 / (-1.0 - (1.0 + (s * ((s * 0.5) + -1.0)))))) ^ c_n) / (0.5 ^ c_n);
	else
		tmp = (exp(s) + 1.0) ^ -c_p;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if c_n < 2

   1. Initial program 95.9%

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

      1. associate-/l/ 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in t around 0 96.5%

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

   6. Taylor expanded in c_p around 0 99.0%

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

   7. Taylor expanded in s around 0 99.3%

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

   if 2 < c_n

   1. Initial program 14.3%

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

      1. associate-/l/ 14.3%

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

   3. Simplified 14.3%

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

   5. Taylor expanded in c_n around 0 37.4%

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

      1. *-un-lft-identity 37.4%

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

      2. inv-pow 37.4%

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

      3. pow-pow 37.4%

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

      4. add-sqr-sqrt 21.5%

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

      5. sqrt-unprod 78.8%

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

      6. sqr-neg 78.8%

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

      7. sqrt-unprod 57.3%

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

      8. add-sqr-sqrt 78.8%

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

   7. Applied egg-rr 78.8%

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

      1. *-lft-identity 78.8%

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

      2. neg-mul-1 78.8%

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

   9. Simplified 78.8%

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

   10. Taylor expanded in c_p around 0 86.2%

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

4. Final simplification 98.6%

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

Alternative 4: 95.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_n \leq 10^{-58}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{s} + 1\right)}^{\left(-c\_p\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_n 1e-58) 1.0 (pow (+ (exp s) 1.0) (- c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_n <= 1e-58) {
		tmp = 1.0;
	} else {
		tmp = pow((exp(s) + 1.0), -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 (c_n <= 1d-58) then
        tmp = 1.0d0
    else
        tmp = (exp(s) + 1.0d0) ** -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 (c_n <= 1e-58) {
		tmp = 1.0;
	} else {
		tmp = Math.pow((Math.exp(s) + 1.0), -c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_n <= 1e-58:
		tmp = 1.0
	else:
		tmp = math.pow((math.exp(s) + 1.0), -c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_n <= 1e-58)
		tmp = 1.0;
	else
		tmp = Float64(exp(s) + 1.0) ^ Float64(-c_p);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_n <= 1e-58)
		tmp = 1.0;
	else
		tmp = (exp(s) + 1.0) ^ -c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$n, 1e-58], 1.0, N[Power[N[(N[Exp[s], $MachinePrecision] + 1.0), $MachinePrecision], (-c$95$p)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_n < 1e-58

   1. Initial program 96.3%

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

      1. associate-/l/ 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in c_n around 0 97.4%

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

   6. Taylor expanded in c_p around 0 99.5%

      \[\leadsto \color{blue}{1} \]

   if 1e-58 < c_n

   1. Initial program 77.8%

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

      1. associate-/l/ 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in c_n around 0 82.5%

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

      1. *-un-lft-identity 82.5%

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

      2. inv-pow 82.5%

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

      3. pow-pow 82.5%

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

      4. add-sqr-sqrt 36.0%

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

      5. sqrt-unprod 91.1%

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

      6. sqr-neg 91.1%

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

      7. sqrt-unprod 55.1%

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

      8. add-sqr-sqrt 92.8%

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

   7. Applied egg-rr 92.8%

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

      1. *-lft-identity 92.8%

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

      2. neg-mul-1 92.8%

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

   9. Simplified 92.8%

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

   10. Taylor expanded in c_p around 0 95.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_n \leq 10^{-58}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{s} + 1\right)}^{\left(-c\_p\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 91.5%

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

   1. associate-/l/ 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in c_n around 0 93.5%

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

6. Taylor expanded in c_p around 0 95.7%

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

Developer target: 96.8% 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 2024110 
(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))))