Harley's example

Percentage Accurate: 91.8% → 97.0%

Time: 1.3min

Alternatives: 3

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: 91.8% 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: 97.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\left(\frac{1 + e^{-s}}{2}\right)}^{c\_p}} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (/ 1.0 (pow (/ (+ 1.0 (exp (- s))) 2.0) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	return 1.0 / pow(((1.0 + exp(-s)) / 2.0), 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 / (((1.0d0 + exp(-s)) / 2.0d0) ** c_p)
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	return 1.0 / Math.pow(((1.0 + Math.exp(-s)) / 2.0), c_p);
}

Python

def code(c_p, c_n, t, s):
	return 1.0 / math.pow(((1.0 + math.exp(-s)) / 2.0), c_p)

Julia

function code(c_p, c_n, t, s)
	return Float64(1.0 / (Float64(Float64(1.0 + exp(Float64(-s))) / 2.0) ^ c_p))
end

MATLAB

function tmp = code(c_p, c_n, t, s)
	tmp = 1.0 / (((1.0 + exp(-s)) / 2.0) ^ c_p);
end

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-/l* 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in c_n around 0 94.0%

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

   1. div-inv 94.0%

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

   2. inv-pow 94.0%

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

   3. pow-pow 94.0%

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

   4. pow-flip 94.0%

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

7. Applied egg-rr 94.0%

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

   1. neg-mul-1 94.0%

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

9. Simplified 94.0%

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

    1. pow-prod-down 97.5%

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

    2. pow-neg 97.5%

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

    3. add-exp-log 97.5%

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

    4. add-exp-log 97.5%

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

    5. prod-exp 97.5%

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

    6. log1p-define 97.5%

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

    7. log-rec 97.5%

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

    8. log1p-define 97.5%

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

11. Applied egg-rr 97.5%

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

    1. log1p-define 97.5%

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

    2. log1p-undefine 97.5%

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

    3. log-rec 97.5%

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

    4. exp-sum 97.5%

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

    5. rem-exp-log 97.5%

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

    6. rem-exp-log 97.5%

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

    7. associate-*r/ 97.5%

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

    8. *-commutative 97.5%

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

    9. *-lft-identity 97.5%

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

13. Simplified 97.5%

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

14. Taylor expanded in t around 0 97.6%

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

15. Final simplification 97.6%

    \[\leadsto \frac{1}{{\left(\frac{1 + e^{-s}}{2}\right)}^{c\_p}} \]
16. Add Preprocessing

Alternative 2: 95.2% accurate, 119.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-/l* 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in c_n around 0 94.0%

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

6. Taylor expanded in s around 0 92.1%

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

7. Taylor expanded in t around 0 95.0%

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

   1. *-commutative 95.0%

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

9. Simplified 95.0%

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

10. Final simplification 95.0%

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

Alternative 3: 95.1% accurate, 835.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-/l* 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in c_n around 0 94.0%

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

6. Taylor expanded in c_p around 0 94.9%

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

7. Final simplification 94.9%

   \[\leadsto 1 \]
8. Add Preprocessing

Developer target: 97.7% 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 2024046 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

  :herbie-target
  (* (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))))