Harley's example

Percentage Accurate: 91.1% → 98.2%

Time: 2.4min

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: 91.1% 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: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-t}\\ t_2 := c\_p \cdot \mathsf{log1p}\left(t\_1\right)\\ t_3 := e^{-s}\\ \mathbf{if}\;-t \leq -5.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{{0.5}^{c\_p}}{1 - t\_2}\\ \mathbf{elif}\;-t \leq 100:\\ \;\;\;\;e^{t\_2 + \left(\left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - t\_3}\right) - c\_p \cdot \mathsf{log1p}\left(t\_3\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - t\_1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_n}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- t))) (t_2 (* c_p (log1p t_1))) (t_3 (exp (- s))))
   (if (<= (- t) -5.2e-44)
     (/ (pow 0.5 c_p) (- 1.0 t_2))
     (if (<= (- t) 100.0)
       (exp
        (+
         t_2
         (-
          (- (* c_n (log1p (/ 1.0 (- -1.0 t_3)))) (* c_p (log1p t_3)))
          (* c_n (log1p (/ 1.0 (- -1.0 t_1)))))))
       (pow 0.5 c_n)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-t);
	double t_2 = c_p * log1p(t_1);
	double t_3 = exp(-s);
	double tmp;
	if (-t <= -5.2e-44) {
		tmp = pow(0.5, c_p) / (1.0 - t_2);
	} else if (-t <= 100.0) {
		tmp = exp((t_2 + (((c_n * log1p((1.0 / (-1.0 - t_3)))) - (c_p * log1p(t_3))) - (c_n * log1p((1.0 / (-1.0 - t_1)))))));
	} else {
		tmp = pow(0.5, c_n);
	}
	return tmp;
}

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = Math.exp(-t);
	double t_2 = c_p * Math.log1p(t_1);
	double t_3 = Math.exp(-s);
	double tmp;
	if (-t <= -5.2e-44) {
		tmp = Math.pow(0.5, c_p) / (1.0 - t_2);
	} else if (-t <= 100.0) {
		tmp = Math.exp((t_2 + (((c_n * Math.log1p((1.0 / (-1.0 - t_3)))) - (c_p * Math.log1p(t_3))) - (c_n * Math.log1p((1.0 / (-1.0 - t_1)))))));
	} else {
		tmp = Math.pow(0.5, c_n);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	t_1 = math.exp(-t)
	t_2 = c_p * math.log1p(t_1)
	t_3 = math.exp(-s)
	tmp = 0
	if -t <= -5.2e-44:
		tmp = math.pow(0.5, c_p) / (1.0 - t_2)
	elif -t <= 100.0:
		tmp = math.exp((t_2 + (((c_n * math.log1p((1.0 / (-1.0 - t_3)))) - (c_p * math.log1p(t_3))) - (c_n * math.log1p((1.0 / (-1.0 - t_1)))))))
	else:
		tmp = math.pow(0.5, c_n)
	return tmp

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-t))
	t_2 = Float64(c_p * log1p(t_1))
	t_3 = exp(Float64(-s))
	tmp = 0.0
	if (Float64(-t) <= -5.2e-44)
		tmp = Float64((0.5 ^ c_p) / Float64(1.0 - t_2));
	elseif (Float64(-t) <= 100.0)
		tmp = exp(Float64(t_2 + Float64(Float64(Float64(c_n * log1p(Float64(1.0 / Float64(-1.0 - t_3)))) - Float64(c_p * log1p(t_3))) - Float64(c_n * log1p(Float64(1.0 / Float64(-1.0 - t_1)))))));
	else
		tmp = 0.5 ^ c_n;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-t)], $MachinePrecision]}, Block[{t$95$2 = N[(c$95$p * N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Exp[(-s)], $MachinePrecision]}, If[LessEqual[(-t), -5.2e-44], N[(N[Power[0.5, c$95$p], $MachinePrecision] / N[(1.0 - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[(-t), 100.0], N[Exp[N[(t$95$2 + N[(N[(N[(c$95$n * N[Log[1 + N[(1.0 / N[(-1.0 - t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(c$95$p * N[Log[1 + t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c$95$n * N[Log[1 + N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[0.5, c$95$n], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (neg.f64 t) < -5.1999999999999996e-44

   1. Initial program 77.7%

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

      1. associate-/l* 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in c_n around 0 95.6%

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

   6. Taylor expanded in c_p around 0 95.6%

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

      1. log-rec 95.6%

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

      2. log1p-undefine 95.6%

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

      3. distribute-rgt-neg-out 95.6%

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

      4. unsub-neg 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in s around 0 100.0%

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

   if -5.1999999999999996e-44 < (neg.f64 t) < 100

   1. Initial program 91.9%

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

      1. associate-/l/ 91.9%

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

   3. Simplified 91.9%

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

      1. add-exp-log 91.9%

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

      2. log-div 91.9%

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

   6. Applied egg-rr 99.4%

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

   if 100 < (neg.f64 t)

   1. Initial program 1.3%

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

      1. associate-/l* 1.3%

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

   3. Simplified 1.3%

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

   5. Taylor expanded in c_p around 0 100.0%

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

   6. Taylor expanded in s around 0 100.0%

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

   7. Taylor expanded in t around 0 50.0%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(1 - \frac{1}{\color{blue}{2 + t \cdot \left(0.5 \cdot t - 1\right)}}\right)}^{c\_n}} \]

   8. Taylor expanded in t around inf 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-t \leq -5.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{{0.5}^{c\_p}}{1 - c\_p \cdot \mathsf{log1p}\left(e^{-t}\right)}\\ \mathbf{elif}\;-t \leq 100:\\ \;\;\;\;e^{c\_p \cdot \mathsf{log1p}\left(e^{-t}\right) + \left(\left(c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - e^{-s}}\right) - c\_p \cdot \mathsf{log1p}\left(e^{-s}\right)\right) - c\_n \cdot \mathsf{log1p}\left(\frac{1}{-1 - e^{-t}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{0.5}^{c\_n}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.84:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{1}{1 + e^{-s}}}{0.5}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= t -0.84)
   (/ (pow 0.5 c_n) (pow (+ 1.0 (/ 1.0 (- -1.0 (exp (- t))))) c_n))
   (pow (/ (/ 1.0 (+ 1.0 (exp (- s)))) 0.5) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -0.84) {
		tmp = pow(0.5, c_n) / pow((1.0 + (1.0 / (-1.0 - exp(-t)))), c_n);
	} else {
		tmp = pow(((1.0 / (1.0 + exp(-s))) / 0.5), 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 (t <= (-0.84d0)) then
        tmp = (0.5d0 ** c_n) / ((1.0d0 + (1.0d0 / ((-1.0d0) - exp(-t)))) ** c_n)
    else
        tmp = ((1.0d0 / (1.0d0 + exp(-s))) / 0.5d0) ** 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 (t <= -0.84) {
		tmp = Math.pow(0.5, c_n) / Math.pow((1.0 + (1.0 / (-1.0 - Math.exp(-t)))), c_n);
	} else {
		tmp = Math.pow(((1.0 / (1.0 + Math.exp(-s))) / 0.5), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if t <= -0.84:
		tmp = math.pow(0.5, c_n) / math.pow((1.0 + (1.0 / (-1.0 - math.exp(-t)))), c_n)
	else:
		tmp = math.pow(((1.0 / (1.0 + math.exp(-s))) / 0.5), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (t <= -0.84)
		tmp = Float64((0.5 ^ c_n) / (Float64(1.0 + Float64(1.0 / Float64(-1.0 - exp(Float64(-t))))) ^ c_n));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + exp(Float64(-s)))) / 0.5) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (t <= -0.84)
		tmp = (0.5 ^ c_n) / ((1.0 + (1.0 / (-1.0 - exp(-t)))) ^ c_n);
	else
		tmp = ((1.0 / (1.0 + exp(-s))) / 0.5) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[t, -0.84], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(1.0 + N[(1.0 / N[(-1.0 - N[Exp[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.839999999999999969

   1. Initial program 37.7%

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

      1. associate-/l* 37.7%

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

   3. Simplified 37.7%

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

   5. Taylor expanded in c_p around 0 92.8%

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

   6. Taylor expanded in s around 0 92.8%

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

   if -0.839999999999999969 < t

   1. Initial program 90.3%

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

      1. associate-/l* 90.3%

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

   3. Simplified 90.3%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}} \]
   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. Taylor expanded in t around 0 93.5%

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

   7. Taylor expanded in s around -inf 93.5%

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

      1. neg-mul-1 93.5%

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

      2. exp-to-pow 93.5%

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

      3. *-commutative 93.5%

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

      4. log-rec 93.5%

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

      5. log1p-undefine 93.5%

         \[\leadsto \frac{e^{c\_p \cdot \left(-\color{blue}{\mathsf{log1p}\left(e^{-s}\right)}\right)}}{{0.5}^{c\_p}} \]

      6. exp-to-pow 93.5%

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

      7. *-commutative 93.5%

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

      8. div-exp 97.4%

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

      9. log1p-undefine 97.4%

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

      10. log-rec 97.4%

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

   9. Simplified 97.4%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.84:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(1 + \frac{1}{-1 - e^{-t}}\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{1}{1 + e^{-s}}}{0.5}\right)}^{c\_p}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.5:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(0.5 + t \cdot -0.25\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{1}{1 + e^{-s}}}{0.5}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= t -0.5)
   (/ (pow 0.5 c_n) (pow (+ 0.5 (* t -0.25)) c_n))
   (pow (/ (/ 1.0 (+ 1.0 (exp (- s)))) 0.5) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -0.5) {
		tmp = pow(0.5, c_n) / pow((0.5 + (t * -0.25)), c_n);
	} else {
		tmp = pow(((1.0 / (1.0 + exp(-s))) / 0.5), 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 (t <= (-0.5d0)) then
        tmp = (0.5d0 ** c_n) / ((0.5d0 + (t * (-0.25d0))) ** c_n)
    else
        tmp = ((1.0d0 / (1.0d0 + exp(-s))) / 0.5d0) ** 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 (t <= -0.5) {
		tmp = Math.pow(0.5, c_n) / Math.pow((0.5 + (t * -0.25)), c_n);
	} else {
		tmp = Math.pow(((1.0 / (1.0 + Math.exp(-s))) / 0.5), c_p);
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if t <= -0.5:
		tmp = math.pow(0.5, c_n) / math.pow((0.5 + (t * -0.25)), c_n)
	else:
		tmp = math.pow(((1.0 / (1.0 + math.exp(-s))) / 0.5), c_p)
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (t <= -0.5)
		tmp = Float64((0.5 ^ c_n) / (Float64(0.5 + Float64(t * -0.25)) ^ c_n));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + exp(Float64(-s)))) / 0.5) ^ c_p;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (t <= -0.5)
		tmp = (0.5 ^ c_n) / ((0.5 + (t * -0.25)) ^ c_n);
	else
		tmp = ((1.0 / (1.0 + exp(-s))) / 0.5) ^ c_p;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[t, -0.5], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(0.5 + N[(t * -0.25), $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(1.0 / N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 0.5), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.5

   1. Initial program 37.7%

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

      1. associate-/l* 37.7%

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

   3. Simplified 37.7%

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

   5. Taylor expanded in c_p around 0 92.8%

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

   6. Taylor expanded in s around 0 92.8%

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

   7. Taylor expanded in t around 0 90.6%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\color{blue}{\left(0.5 + -0.25 \cdot t\right)}}^{c\_n}} \]
   8. Step-by-step derivation

      1. *-commutative 90.6%

         \[\leadsto \frac{{0.5}^{c\_n}}{{\left(0.5 + \color{blue}{t \cdot -0.25}\right)}^{c\_n}} \]

   9. Simplified 90.6%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\color{blue}{\left(0.5 + t \cdot -0.25\right)}}^{c\_n}} \]

   if -0.5 < t

   1. Initial program 90.3%

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

      1. associate-/l* 90.3%

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

   3. Simplified 90.3%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot \frac{{\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}}} \]
   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. Taylor expanded in t around 0 93.5%

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

   7. Taylor expanded in s around -inf 93.5%

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

      1. neg-mul-1 93.5%

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

      2. exp-to-pow 93.5%

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

      3. *-commutative 93.5%

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

      4. log-rec 93.5%

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

      5. log1p-undefine 93.5%

         \[\leadsto \frac{e^{c\_p \cdot \left(-\color{blue}{\mathsf{log1p}\left(e^{-s}\right)}\right)}}{{0.5}^{c\_p}} \]

      6. exp-to-pow 93.5%

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

      7. *-commutative 93.5%

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

      8. div-exp 97.4%

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

      9. log1p-undefine 97.4%

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

      10. log-rec 97.4%

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

   9. Simplified 97.4%

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

Alternative 4: 95.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-82}:\\ \;\;\;\;{0.5}^{c\_n}\\ \mathbf{else}:\\ \;\;\;\;1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(c\_p \cdot -0.125 + 0.125 \cdot {c\_p}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= t -2e-82)
   (pow 0.5 c_n)
   (+
    1.0
    (* s (+ (* 0.5 c_p) (* s (+ (* c_p -0.125) (* 0.125 (pow c_p 2.0)))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -2e-82) {
		tmp = pow(0.5, c_n);
	} else {
		tmp = 1.0 + (s * ((0.5 * c_p) + (s * ((c_p * -0.125) + (0.125 * pow(c_p, 2.0))))));
	}
	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 (t <= (-2d-82)) then
        tmp = 0.5d0 ** c_n
    else
        tmp = 1.0d0 + (s * ((0.5d0 * c_p) + (s * ((c_p * (-0.125d0)) + (0.125d0 * (c_p ** 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -2e-82) {
		tmp = Math.pow(0.5, c_n);
	} else {
		tmp = 1.0 + (s * ((0.5 * c_p) + (s * ((c_p * -0.125) + (0.125 * Math.pow(c_p, 2.0))))));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if t <= -2e-82:
		tmp = math.pow(0.5, c_n)
	else:
		tmp = 1.0 + (s * ((0.5 * c_p) + (s * ((c_p * -0.125) + (0.125 * math.pow(c_p, 2.0))))))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (t <= -2e-82)
		tmp = 0.5 ^ c_n;
	else
		tmp = Float64(1.0 + Float64(s * Float64(Float64(0.5 * c_p) + Float64(s * Float64(Float64(c_p * -0.125) + Float64(0.125 * (c_p ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (t <= -2e-82)
		tmp = 0.5 ^ c_n;
	else
		tmp = 1.0 + (s * ((0.5 * c_p) + (s * ((c_p * -0.125) + (0.125 * (c_p ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[t, -2e-82], N[Power[0.5, c$95$n], $MachinePrecision], N[(1.0 + N[(s * N[(N[(0.5 * c$95$p), $MachinePrecision] + N[(s * N[(N[(c$95$p * -0.125), $MachinePrecision] + N[(0.125 * N[Power[c$95$p, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2e-82

   1. Initial program 70.5%

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

      1. associate-/l* 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in c_p around 0 86.7%

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

   6. Taylor expanded in s around 0 86.7%

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

   7. Taylor expanded in t around 0 73.2%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(1 - \frac{1}{\color{blue}{2 + t \cdot \left(0.5 \cdot t - 1\right)}}\right)}^{c\_n}} \]

   8. Taylor expanded in t around inf 86.7%

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

   if -2e-82 < t

   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%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in c_n around 0 94.6%

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

   6. Taylor expanded in t around 0 93.6%

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

   7. Taylor expanded in s around 0 95.0%

      \[\leadsto \color{blue}{1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_p + 0.125 \cdot {c\_p}^{2}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-82}:\\ \;\;\;\;{0.5}^{c\_n}\\ \mathbf{else}:\\ \;\;\;\;1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(c\_p \cdot -0.125 + 0.125 \cdot {c\_p}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-82}:\\ \;\;\;\;{0.5}^{c\_n}\\ \mathbf{else}:\\ \;\;\;\;1 + s \cdot \left(0.5 \cdot c\_p\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= t -2.8e-82) (pow 0.5 c_n) (+ 1.0 (* s (* 0.5 c_p)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -2.8e-82) {
		tmp = pow(0.5, c_n);
	} else {
		tmp = 1.0 + (s * (0.5 * 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 (t <= (-2.8d-82)) then
        tmp = 0.5d0 ** c_n
    else
        tmp = 1.0d0 + (s * (0.5d0 * 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 (t <= -2.8e-82) {
		tmp = Math.pow(0.5, c_n);
	} else {
		tmp = 1.0 + (s * (0.5 * c_p));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if t <= -2.8e-82:
		tmp = math.pow(0.5, c_n)
	else:
		tmp = 1.0 + (s * (0.5 * c_p))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (t <= -2.8e-82)
		tmp = 0.5 ^ c_n;
	else
		tmp = Float64(1.0 + Float64(s * Float64(0.5 * c_p)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (t <= -2.8e-82)
		tmp = 0.5 ^ c_n;
	else
		tmp = 1.0 + (s * (0.5 * c_p));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[t, -2.8e-82], N[Power[0.5, c$95$n], $MachinePrecision], N[(1.0 + N[(s * N[(0.5 * c$95$p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.80000000000000024e-82

   1. Initial program 70.5%

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

      1. associate-/l* 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in c_p around 0 86.7%

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

   6. Taylor expanded in s around 0 86.7%

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

   7. Taylor expanded in t around 0 73.2%

      \[\leadsto \frac{{0.5}^{c\_n}}{{\left(1 - \frac{1}{\color{blue}{2 + t \cdot \left(0.5 \cdot t - 1\right)}}\right)}^{c\_n}} \]

   8. Taylor expanded in t around inf 86.7%

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

   if -2.80000000000000024e-82 < t

   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%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in c_n around 0 94.6%

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

   6. Taylor expanded in t around 0 93.6%

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

   7. Taylor expanded in s around 0 94.9%

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

      1. associate-*r* 94.9%

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

   9. Simplified 94.9%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-82}:\\ \;\;\;\;{0.5}^{c\_n}\\ \mathbf{else}:\\ \;\;\;\;1 + s \cdot \left(0.5 \cdot c\_p\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

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 + (s * (0.5d0 * c_p))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

   1. associate-/l* 86.4%

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

3. Simplified 86.4%

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

5. Taylor expanded in c_n around 0 89.8%

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

6. Taylor expanded in t around 0 90.6%

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

7. Taylor expanded in s around 0 90.6%

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

   1. associate-*r* 90.6%

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

9. Simplified 90.6%

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

10. Final simplification 90.6%

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

Alternative 7: 94.3% 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 86.4%

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

   1. associate-/l* 86.4%

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

3. Simplified 86.4%

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

5. Taylor expanded in c_p around 0 90.0%

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

6. Taylor expanded in s around 0 90.4%

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

7. Taylor expanded in c_n around 0 90.5%

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

Developer Target 1: 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 2024151 
(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))))