Harley's example

Percentage Accurate: 90.8% → 99.5%

Time: 1.2min

Alternatives: 12

Speedup: 835.0×

• could not determine a ground truth

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

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.5 \cdot c\_n + 0.5 \cdot c\_p\\ e^{s \cdot t\_1 - t \cdot \left(t\_1 + t \cdot \left({t}^{2} \cdot \left(c\_n \cdot 0.005208333333333333 + c\_p \cdot 0.005208333333333333\right) + \left(c\_n \cdot -0.125 + c\_p \cdot -0.125\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (+ (* -0.5 c_n) (* 0.5 c_p))))
   (exp
    (-
     (* s t_1)
     (*
      t
      (+
       t_1
       (*
        t
        (+
         (*
          (pow t 2.0)
          (+ (* c_n 0.005208333333333333) (* c_p 0.005208333333333333)))
         (+ (* c_n -0.125) (* c_p -0.125))))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = (-0.5 * c_n) + (0.5 * c_p);
	return exp(((s * t_1) - (t * (t_1 + (t * ((pow(t, 2.0) * ((c_n * 0.005208333333333333) + (c_p * 0.005208333333333333))) + ((c_n * -0.125) + (c_p * -0.125))))))));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: t_1
    t_1 = ((-0.5d0) * c_n) + (0.5d0 * c_p)
    code = exp(((s * t_1) - (t * (t_1 + (t * (((t ** 2.0d0) * ((c_n * 0.005208333333333333d0) + (c_p * 0.005208333333333333d0))) + ((c_n * (-0.125d0)) + (c_p * (-0.125d0)))))))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = (-0.5 * c_n) + (0.5 * c_p);
	return Math.exp(((s * t_1) - (t * (t_1 + (t * ((Math.pow(t, 2.0) * ((c_n * 0.005208333333333333) + (c_p * 0.005208333333333333))) + ((c_n * -0.125) + (c_p * -0.125))))))));
}

Python

def code(c_p, c_n, t, s):
	t_1 = (-0.5 * c_n) + (0.5 * c_p)
	return math.exp(((s * t_1) - (t * (t_1 + (t * ((math.pow(t, 2.0) * ((c_n * 0.005208333333333333) + (c_p * 0.005208333333333333))) + ((c_n * -0.125) + (c_p * -0.125))))))))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(Float64(-0.5 * c_n) + Float64(0.5 * c_p))
	return exp(Float64(Float64(s * t_1) - Float64(t * Float64(t_1 + Float64(t * Float64(Float64((t ^ 2.0) * Float64(Float64(c_n * 0.005208333333333333) + Float64(c_p * 0.005208333333333333))) + Float64(Float64(c_n * -0.125) + Float64(c_p * -0.125))))))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = (-0.5 * c_n) + (0.5 * c_p);
	tmp = exp(((s * t_1) - (t * (t_1 + (t * (((t ^ 2.0) * ((c_n * 0.005208333333333333) + (c_p * 0.005208333333333333))) + ((c_n * -0.125) + (c_p * -0.125))))))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[(-0.5 * c$95$n), $MachinePrecision] + N[(0.5 * c$95$p), $MachinePrecision]), $MachinePrecision]}, N[Exp[N[(N[(s * t$95$1), $MachinePrecision] - N[(t * N[(t$95$1 + N[(t * N[(N[(N[Power[t, 2.0], $MachinePrecision] * N[(N[(c$95$n * 0.005208333333333333), $MachinePrecision] + N[(c$95$p * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c$95$n * -0.125), $MachinePrecision] + N[(c$95$p * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.4%

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

3. Simplified 91.4%

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

   1. add-exp-log 91.4%

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

   2. log-div 91.5%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in s around 0 96.7%

   \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

8. Taylor expanded in t around 0 99.8%

   \[\leadsto e^{\color{blue}{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(0.005208333333333333 \cdot c\_n + 0.005208333333333333 \cdot c\_p\right)\right) - \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right) - \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)}} \]

9. Final simplification 99.8%

   \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) - t \cdot \left(\left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + t \cdot \left({t}^{2} \cdot \left(c\_n \cdot 0.005208333333333333 + c\_p \cdot 0.005208333333333333\right) + \left(c\_n \cdot -0.125 + c\_p \cdot -0.125\right)\right)\right)} \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(c\_n \cdot t\right) \cdot \left(0.5 + t \cdot \left({t}^{2} \cdot -0.005208333333333333 + 0.125\right)\right)} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (exp
  (+
   (* s (+ (* -0.5 c_n) (* 0.5 c_p)))
   (*
    (* c_n t)
    (+ 0.5 (* t (+ (* (pow t 2.0) -0.005208333333333333) 0.125)))))))

C

double code(double c_p, double c_n, double t, double s) {
	return exp(((s * ((-0.5 * c_n) + (0.5 * c_p))) + ((c_n * t) * (0.5 + (t * ((pow(t, 2.0) * -0.005208333333333333) + 0.125))))));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = exp(((s * (((-0.5d0) * c_n) + (0.5d0 * c_p))) + ((c_n * t) * (0.5d0 + (t * (((t ** 2.0d0) * (-0.005208333333333333d0)) + 0.125d0))))))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return Math.exp(((s * ((-0.5 * c_n) + (0.5 * c_p))) + ((c_n * t) * (0.5 + (t * ((Math.pow(t, 2.0) * -0.005208333333333333) + 0.125))))));
}

Python

def code(c_p, c_n, t, s):
	return math.exp(((s * ((-0.5 * c_n) + (0.5 * c_p))) + ((c_n * t) * (0.5 + (t * ((math.pow(t, 2.0) * -0.005208333333333333) + 0.125))))))

Julia

function code(c_p, c_n, t, s)
	return exp(Float64(Float64(s * Float64(Float64(-0.5 * c_n) + Float64(0.5 * c_p))) + Float64(Float64(c_n * t) * Float64(0.5 + Float64(t * Float64(Float64((t ^ 2.0) * -0.005208333333333333) + 0.125))))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = exp(((s * ((-0.5 * c_n) + (0.5 * c_p))) + ((c_n * t) * (0.5 + (t * (((t ^ 2.0) * -0.005208333333333333) + 0.125))))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[Exp[N[(N[(s * N[(N[(-0.5 * c$95$n), $MachinePrecision] + N[(0.5 * c$95$p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c$95$n * t), $MachinePrecision] * N[(0.5 + N[(t * N[(N[(N[Power[t, 2.0], $MachinePrecision] * -0.005208333333333333), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.4%

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

3. Simplified 91.4%

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

   1. add-exp-log 91.4%

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

   2. log-div 91.5%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in s around 0 96.7%

   \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

8. Taylor expanded in t around 0 99.8%

   \[\leadsto e^{\color{blue}{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(0.005208333333333333 \cdot c\_n + 0.005208333333333333 \cdot c\_p\right)\right) - \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right) - \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)}} \]

9. Taylor expanded in c_n around inf 99.7%

   \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{c\_n \cdot \left(t \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r* 99.7%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(c\_n \cdot t\right) \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)}} \]

    2. *-commutative 99.7%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right)} \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)} \]

    3. +-commutative 99.7%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \color{blue}{\left(-0.005208333333333333 \cdot {t}^{2} + 0.125\right)}\right)} \]

    4. *-commutative 99.7%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left(\color{blue}{{t}^{2} \cdot -0.005208333333333333} + 0.125\right)\right)} \]

11. Simplified 99.7%

    \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left({t}^{2} \cdot -0.005208333333333333 + 0.125\right)\right)}} \]

12. Final simplification 99.7%

    \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(c\_n \cdot t\right) \cdot \left(0.5 + t \cdot \left({t}^{2} \cdot -0.005208333333333333 + 0.125\right)\right)} \]
13. Add Preprocessing

Alternative 3: 99.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.5 \cdot c\_n + 0.5 \cdot c\_p\\ e^{s \cdot t\_1 - t\_1 \cdot t} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (+ (* -0.5 c_n) (* 0.5 c_p)))) (exp (- (* s t_1) (* t_1 t)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = (-0.5 * c_n) + (0.5 * c_p);
	return exp(((s * t_1) - (t_1 * 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
    real(8) :: t_1
    t_1 = ((-0.5d0) * c_n) + (0.5d0 * c_p)
    code = exp(((s * t_1) - (t_1 * t)))
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double t_1 = (-0.5 * c_n) + (0.5 * c_p);
	return Math.exp(((s * t_1) - (t_1 * t)));
}

Python

def code(c_p, c_n, t, s):
	t_1 = (-0.5 * c_n) + (0.5 * c_p)
	return math.exp(((s * t_1) - (t_1 * t)))

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(Float64(-0.5 * c_n) + Float64(0.5 * c_p))
	return exp(Float64(Float64(s * t_1) - Float64(t_1 * t)))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	t_1 = (-0.5 * c_n) + (0.5 * c_p);
	tmp = exp(((s * t_1) - (t_1 * t)));
end

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.4%

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

3. Simplified 91.4%

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

   1. add-exp-log 91.4%

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

   2. log-div 91.5%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in s around 0 96.7%

   \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

8. Taylor expanded in t around 0 99.7%

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

9. Final simplification 99.7%

   \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) - \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) \cdot t} \]
10. Add Preprocessing

Alternative 4: 99.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -3 \cdot 10^{+21}:\\ \;\;\;\;e^{c\_p \cdot \left(s \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(c\_n \cdot 0.5\right) \cdot \left(t - s\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -3e+21) (exp (* c_p (* s 0.5))) (exp (* (* c_n 0.5) (- t s)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -3e+21) {
		tmp = exp((c_p * (s * 0.5)));
	} else {
		tmp = exp(((c_n * 0.5) * (t - s)));
	}
	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 <= (-3d+21)) then
        tmp = exp((c_p * (s * 0.5d0)))
    else
        tmp = exp(((c_n * 0.5d0) * (t - s)))
    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 <= -3e+21) {
		tmp = Math.exp((c_p * (s * 0.5)));
	} else {
		tmp = Math.exp(((c_n * 0.5) * (t - s)));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -3e+21:
		tmp = math.exp((c_p * (s * 0.5)))
	else:
		tmp = math.exp(((c_n * 0.5) * (t - s)))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -3e+21)
		tmp = exp(Float64(c_p * Float64(s * 0.5)));
	else
		tmp = exp(Float64(Float64(c_n * 0.5) * Float64(t - s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -3e+21)
		tmp = exp((c_p * (s * 0.5)));
	else
		tmp = exp(((c_n * 0.5) * (t - s)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -3e+21], N[Exp[N[(c$95$p * N[(s * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(c$95$n * 0.5), $MachinePrecision] * N[(t - s), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -3e21

   1. Initial program 80.0%

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

      1. associate-/l/ 80.0%

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

   3. Simplified 80.0%

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

      1. add-exp-log 80.0%

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

      2. log-div 80.0%

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

   6. Applied egg-rr 80.0%

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

   7. Taylor expanded in s around 0 80.0%

      \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   8. Taylor expanded in t around 0 100.0%

      \[\leadsto e^{\color{blue}{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(0.005208333333333333 \cdot c\_n + 0.005208333333333333 \cdot c\_p\right)\right) - \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right) - \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)}} \]

   9. Taylor expanded in c_n around inf 100.0%

      \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{c\_n \cdot \left(t \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(c\_n \cdot t\right) \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)}} \]

       2. *-commutative 100.0%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right)} \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)} \]

       3. +-commutative 100.0%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \color{blue}{\left(-0.005208333333333333 \cdot {t}^{2} + 0.125\right)}\right)} \]

       4. *-commutative 100.0%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left(\color{blue}{{t}^{2} \cdot -0.005208333333333333} + 0.125\right)\right)} \]

   11. Simplified 100.0%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left({t}^{2} \cdot -0.005208333333333333 + 0.125\right)\right)}} \]

   12. Taylor expanded in c_n around 0 100.0%

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

       1. *-commutative 100.0%

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

       2. associate-*r* 100.0%

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

   14. Simplified 100.0%

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

   if -3e21 < s

   1. Initial program 91.7%

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

      1. associate-/l/ 91.7%

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

   3. Simplified 91.7%

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

      1. add-exp-log 91.7%

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

      2. log-div 91.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in s around 0 97.0%

      \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   8. Taylor expanded in t around 0 99.7%

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

   9. Taylor expanded in c_n around inf 99.1%

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

   10. Taylor expanded in s around 0 99.1%

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

       1. +-commutative 99.1%

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

       2. *-commutative 99.1%

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

       3. *-commutative 99.1%

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

       4. associate-*r* 99.1%

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

       5. associate-*r* 99.1%

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

       6. *-commutative 99.1%

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

       7. metadata-eval 99.1%

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

       8. distribute-rgt-neg-in 99.1%

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

       9. cancel-sign-sub-inv 99.1%

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

       10. *-commutative 99.1%

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

       11. distribute-rgt-out-- 99.1%

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

       12. *-commutative 99.1%

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

   12. Simplified 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq -3 \cdot 10^{+21}:\\ \;\;\;\;e^{c\_p \cdot \left(s \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(c\_n \cdot 0.5\right) \cdot \left(t - s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 10^{-56}:\\ \;\;\;\;e^{s \cdot \left(-0.5 \cdot c\_n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(s \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 1e-56) (exp (* s (* -0.5 c_n))) (exp (* c_p (* s 0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 1e-56) {
		tmp = exp((s * (-0.5 * c_n)));
	} else {
		tmp = exp((c_p * (s * 0.5)));
	}
	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_p <= 1d-56) then
        tmp = exp((s * ((-0.5d0) * c_n)))
    else
        tmp = exp((c_p * (s * 0.5d0)))
    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_p <= 1e-56) {
		tmp = Math.exp((s * (-0.5 * c_n)));
	} else {
		tmp = Math.exp((c_p * (s * 0.5)));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if c_p <= 1e-56:
		tmp = math.exp((s * (-0.5 * c_n)))
	else:
		tmp = math.exp((c_p * (s * 0.5)))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 1e-56)
		tmp = exp(Float64(s * Float64(-0.5 * c_n)));
	else
		tmp = exp(Float64(c_p * Float64(s * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (c_p <= 1e-56)
		tmp = exp((s * (-0.5 * c_n)));
	else
		tmp = exp((c_p * (s * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 1e-56], N[Exp[N[(s * N[(-0.5 * c$95$n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(c$95$p * N[(s * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 1e-56

   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. Step-by-step derivation

      1. add-exp-log 93.1%

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

      2. log-div 93.1%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in s around 0 96.9%

      \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   8. Taylor expanded in s around inf 98.7%

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

   9. Taylor expanded in c_n around inf 98.7%

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

       1. associate-*r* 98.7%

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

       2. *-commutative 98.7%

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

   11. Simplified 98.7%

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

   if 1e-56 < c_p

   1. Initial program 85.2%

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

      1. associate-/l/ 85.2%

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

   3. Simplified 85.2%

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

      1. add-exp-log 85.2%

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

      2. log-div 85.2%

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

   6. Applied egg-rr 96.3%

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

   7. Taylor expanded in s around 0 96.1%

      \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   8. Taylor expanded in t around 0 99.7%

      \[\leadsto e^{\color{blue}{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(0.005208333333333333 \cdot c\_n + 0.005208333333333333 \cdot c\_p\right)\right) - \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right) - \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)}} \]

   9. Taylor expanded in c_n around inf 99.6%

      \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{c\_n \cdot \left(t \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 99.6%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(c\_n \cdot t\right) \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)}} \]

       2. *-commutative 99.6%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right)} \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)} \]

       3. +-commutative 99.6%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \color{blue}{\left(-0.005208333333333333 \cdot {t}^{2} + 0.125\right)}\right)} \]

       4. *-commutative 99.6%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left(\color{blue}{{t}^{2} \cdot -0.005208333333333333} + 0.125\right)\right)} \]

   11. Simplified 99.6%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left({t}^{2} \cdot -0.005208333333333333 + 0.125\right)\right)}} \]

   12. Taylor expanded in c_n around 0 97.8%

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

       1. *-commutative 97.8%

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

       2. associate-*r* 97.8%

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

   14. Simplified 97.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 10^{-56}:\\ \;\;\;\;e^{s \cdot \left(-0.5 \cdot c\_n\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(s \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -360:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(s \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= t -360.0) (exp (* t (* c_n 0.5))) (exp (* c_p (* s 0.5)))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -360.0) {
		tmp = exp((t * (c_n * 0.5)));
	} else {
		tmp = exp((c_p * (s * 0.5)));
	}
	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 <= (-360.0d0)) then
        tmp = exp((t * (c_n * 0.5d0)))
    else
        tmp = exp((c_p * (s * 0.5d0)))
    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 <= -360.0) {
		tmp = Math.exp((t * (c_n * 0.5)));
	} else {
		tmp = Math.exp((c_p * (s * 0.5)));
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if t <= -360.0:
		tmp = math.exp((t * (c_n * 0.5)))
	else:
		tmp = math.exp((c_p * (s * 0.5)))
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (t <= -360.0)
		tmp = exp(Float64(t * Float64(c_n * 0.5)));
	else
		tmp = exp(Float64(c_p * Float64(s * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (t <= -360.0)
		tmp = exp((t * (c_n * 0.5)));
	else
		tmp = exp((c_p * (s * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[t, -360.0], N[Exp[N[(t * N[(c$95$n * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(c$95$p * N[(s * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -360

   1. Initial program 17.2%

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

      1. associate-/l/ 17.2%

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

   3. Simplified 17.2%

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

      1. add-exp-log 17.2%

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

      2. log-div 17.2%

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

   6. Applied egg-rr 17.7%

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

   7. Taylor expanded in s around 0 18.2%

      \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   8. Taylor expanded in t around 0 100.0%

      \[\leadsto e^{\color{blue}{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(0.005208333333333333 \cdot c\_n + 0.005208333333333333 \cdot c\_p\right)\right) - \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right) - \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)}} \]

   9. Taylor expanded in c_n around inf 100.0%

      \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{c\_n \cdot \left(t \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 100.0%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(c\_n \cdot t\right) \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)}} \]

       2. *-commutative 100.0%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right)} \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)} \]

       3. +-commutative 100.0%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \color{blue}{\left(-0.005208333333333333 \cdot {t}^{2} + 0.125\right)}\right)} \]

       4. *-commutative 100.0%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left(\color{blue}{{t}^{2} \cdot -0.005208333333333333} + 0.125\right)\right)} \]

   11. Simplified 100.0%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left({t}^{2} \cdot -0.005208333333333333 + 0.125\right)\right)}} \]

   12. Taylor expanded in t around 0 67.2%

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

   13. Taylor expanded in s around 0 18.0%

       \[\leadsto e^{\color{blue}{t \cdot \left(0.125 \cdot \left(c\_n \cdot t\right) + 0.5 \cdot c\_n\right)}} \]

   14. Taylor expanded in t around 0 100.0%

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

   if -360 < t

   1. Initial program 93.2%

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

      1. associate-/l/ 93.2%

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

   3. Simplified 93.2%

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

      1. add-exp-log 93.2%

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

      2. log-div 93.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in s around 0 98.6%

      \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

   8. Taylor expanded in t around 0 99.8%

      \[\leadsto e^{\color{blue}{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(0.005208333333333333 \cdot c\_n + 0.005208333333333333 \cdot c\_p\right)\right) - \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right) - \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)}} \]

   9. Taylor expanded in c_n around inf 99.7%

      \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{c\_n \cdot \left(t \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 99.7%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(c\_n \cdot t\right) \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)}} \]

       2. *-commutative 99.7%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right)} \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)} \]

       3. +-commutative 99.7%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \color{blue}{\left(-0.005208333333333333 \cdot {t}^{2} + 0.125\right)}\right)} \]

       4. *-commutative 99.7%

          \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left(\color{blue}{{t}^{2} \cdot -0.005208333333333333} + 0.125\right)\right)} \]

   11. Simplified 99.7%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left({t}^{2} \cdot -0.005208333333333333 + 0.125\right)\right)}} \]

   12. Taylor expanded in c_n around 0 98.5%

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

       1. *-commutative 98.5%

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

       2. associate-*r* 98.5%

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

   14. Simplified 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -360:\\ \;\;\;\;e^{t \cdot \left(c\_n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{c\_p \cdot \left(s \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 7.7× speedup

Math

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

FPCore

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

C

double code(double c_p, double c_n, double t, double s) {
	return exp((s * ((-0.5 * c_n) + (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 = exp((s * (((-0.5d0) * c_n) + (0.5d0 * c_p))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.4%

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

3. Simplified 91.4%

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

   1. add-exp-log 91.4%

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

   2. log-div 91.5%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in s around 0 96.7%

   \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

8. Taylor expanded in s around inf 98.9%

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

Alternative 8: 97.8% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.4%

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

3. Simplified 91.4%

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

   1. add-exp-log 91.4%

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

   2. log-div 91.5%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in c_p around 0 96.5%

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

8. Taylor expanded in t around 0 96.3%

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

9. Taylor expanded in s around 0 98.4%

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

10. Taylor expanded in c_n around 0 98.4%

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

11. Final simplification 98.4%

    \[\leadsto e^{c\_n \cdot \left(s \cdot \left(s \cdot -0.125 - 0.5\right)\right)} \]
12. Add Preprocessing

Alternative 9: 95.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.4%

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

3. Simplified 91.4%

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

   1. add-exp-log 91.4%

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

   2. log-div 91.5%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in s around 0 96.7%

   \[\leadsto e^{\color{blue}{\left(\left(-1 \cdot \left(c\_p \cdot \log 2\right) + \left(c\_n \cdot \log 0.5 + s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)\right) - c\_n \cdot \log \left(1 - \frac{1}{1 + e^{-t}}\right)\right)} - c\_p \cdot \left(-\mathsf{log1p}\left(e^{-t}\right)\right)} \]

8. Taylor expanded in t around 0 99.8%

   \[\leadsto e^{\color{blue}{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + t \cdot \left(t \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(0.005208333333333333 \cdot c\_n + 0.005208333333333333 \cdot c\_p\right)\right) - \left(-0.125 \cdot c\_n + -0.125 \cdot c\_p\right)\right) - \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right)\right)}} \]

9. Taylor expanded in c_n around inf 99.7%

   \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{c\_n \cdot \left(t \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)\right)}} \]
10. Step-by-step derivation

    1. associate-*r* 99.7%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(c\_n \cdot t\right) \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)}} \]

    2. *-commutative 99.7%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right)} \cdot \left(0.5 + t \cdot \left(0.125 + -0.005208333333333333 \cdot {t}^{2}\right)\right)} \]

    3. +-commutative 99.7%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \color{blue}{\left(-0.005208333333333333 \cdot {t}^{2} + 0.125\right)}\right)} \]

    4. *-commutative 99.7%

       \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left(\color{blue}{{t}^{2} \cdot -0.005208333333333333} + 0.125\right)\right)} \]

11. Simplified 99.7%

    \[\leadsto e^{s \cdot \left(-0.5 \cdot c\_n + 0.5 \cdot c\_p\right) + \color{blue}{\left(t \cdot c\_n\right) \cdot \left(0.5 + t \cdot \left({t}^{2} \cdot -0.005208333333333333 + 0.125\right)\right)}} \]

12. Taylor expanded in t around 0 99.0%

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

13. Taylor expanded in s around 0 94.3%

    \[\leadsto e^{\color{blue}{t \cdot \left(0.125 \cdot \left(c\_n \cdot t\right) + 0.5 \cdot c\_n\right)}} \]

14. Taylor expanded in t around 0 96.1%

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

15. Final simplification 96.1%

    \[\leadsto e^{t \cdot \left(c\_n \cdot 0.5\right)} \]
16. Add Preprocessing

Alternative 10: 94.4% accurate, 49.1× speedup

Math

\[\begin{array}{l} \\ 1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(c\_p \cdot \left(c\_p \cdot 0.125 - 0.125\right)\right)\right) \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (+ 1.0 (* s (+ (* 0.5 c_p) (* s (* c_p (- (* c_p 0.125) 0.125)))))))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (s * ((0.5 * c_p) + (s * (c_p * ((c_p * 0.125) - 0.125)))));
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    code = 1.0d0 + (s * ((0.5d0 * c_p) + (s * (c_p * ((c_p * 0.125d0) - 0.125d0)))))
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) + (s * (c_p * ((c_p * 0.125) - 0.125)))));
}

Python

def code(c_p, c_n, t, s):
	return 1.0 + (s * ((0.5 * c_p) + (s * (c_p * ((c_p * 0.125) - 0.125)))))

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 + Float64(s * Float64(Float64(0.5 * c_p) + Float64(s * Float64(c_p * Float64(Float64(c_p * 0.125) - 0.125))))))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 + (s * ((0.5 * c_p) + (s * (c_p * ((c_p * 0.125) - 0.125)))));
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := N[(1.0 + N[(s * N[(N[(0.5 * c$95$p), $MachinePrecision] + N[(s * N[(c$95$p * N[(N[(c$95$p * 0.125), $MachinePrecision] - 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

   1. associate-/l* 91.4%

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

3. Simplified 91.4%

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

5. Taylor expanded in c_n around 0 93.6%

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

6. Taylor expanded in t around 0 93.5%

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

7. Taylor expanded in s around 0 94.3%

   \[\leadsto \color{blue}{1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(-0.125 \cdot c\_p + 0.125 \cdot {c\_p}^{2}\right)\right)} \]

8. Taylor expanded in c_p around 0 94.3%

   \[\leadsto 1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \color{blue}{\left(c\_p \cdot \left(0.125 \cdot c\_p - 0.125\right)\right)}\right) \]

9. Final simplification 94.3%

   \[\leadsto 1 + s \cdot \left(0.5 \cdot c\_p + s \cdot \left(c\_p \cdot \left(c\_p \cdot 0.125 - 0.125\right)\right)\right) \]
10. Add Preprocessing

Alternative 11: 94.4% accurate, 119.3× speedup

Math

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

FPCore

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

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 + (0.5 * (s * 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 + (0.5d0 * (s * c_p))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l* 91.4%

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

3. Simplified 91.4%

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

5. Taylor expanded in c_n around 0 93.6%

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

6. Taylor expanded in t around 0 93.5%

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

7. Taylor expanded in s around 0 94.3%

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

   1. *-commutative 94.3%

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

9. Simplified 94.3%

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

Alternative 12: 94.4% 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 91.4%

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

   1. associate-/l* 91.4%

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

3. Simplified 91.4%

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

5. Taylor expanded in c_n around 0 93.6%

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

6. Taylor expanded in c_p around 0 94.2%

   \[\leadsto \color{blue}{1} \]
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 2024107 
(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))))