Harley's example

Percentage Accurate: 90.6% → 98.4%

Time: 52.6s

Alternatives: 4

Speedup: 835.0×

• could not determine a ground truth

  1. c_p = 4.1689321239898914e-113
  2. c_n = 7.4932243126084815e-208
  3. t = 1.2699896696629639e+57
  4. s = -3.811727922192297e-264

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: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

4. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{e^{c\_p \cdot \left(\left(0 - \mathsf{log1p}\left(\frac{1}{e^{s}}\right)\right) - \left(0 - \mathsf{log1p}\left(e^{0 - t}\right)\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{s}}}\right) - \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{t}}}\right)\right)}} \]

5. Taylor expanded in s around 0

   \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c\_p, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, t\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(c\_n, \color{blue}{\left(\left(\log \frac{1}{2} + s \cdot \left(s \cdot \left(\frac{1}{192} \cdot {s}^{2} - \frac{1}{8}\right) - \frac{1}{2}\right)\right) - \log \left(1 - \frac{1}{1 + \frac{1}{e^{t}}}\right)\right)}\right)\right)\right) \]

7. Simplified 93.0%

   \[\leadsto e^{c\_p \cdot \left(\left(0 - \mathsf{log1p}\left(\frac{1}{e^{s}}\right)\right) - \left(0 - \mathsf{log1p}\left(e^{0 - t}\right)\right)\right) + c\_n \cdot \color{blue}{\left(s \cdot \left(s \cdot \left(\left(s \cdot s\right) \cdot 0.005208333333333333 + -0.125\right) + -0.5\right) + \left(\log 0.5 - \mathsf{log1p}\left(\frac{-1}{1 + e^{0 - t}}\right)\right)\right)}} \]

8. Taylor expanded in t around 0

   \[\leadsto \color{blue}{e^{c\_n \cdot \left(s \cdot \left(s \cdot \left(\frac{1}{192} \cdot {s}^{2} - \frac{1}{8}\right) - \frac{1}{2}\right)\right) + c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right)}} \]
9. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(c\_n \cdot \left(s \cdot \left(s \cdot \left(\frac{1}{192} \cdot {s}^{2} - \frac{1}{8}\right) - \frac{1}{2}\right)\right) + c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(c\_n \cdot \left(s \cdot \left(s \cdot \left(\frac{1}{192} \cdot {s}^{2} - \frac{1}{8}\right) - \frac{1}{2}\right)\right)\right), \left(c\_p \cdot \left(\log 2 - \log \left(1 + \frac{1}{e^{s}}\right)\right)\right)\right)\right) \]

10. Simplified 94.3%

    \[\leadsto \color{blue}{e^{c\_n \cdot \left(s \cdot \left(-0.5 + s \cdot \left(-0.125 + s \cdot \left(s \cdot 0.005208333333333333\right)\right)\right)\right) + c\_p \cdot \left(\log 2 - \mathsf{log1p}\left(e^{0 - s}\right)\right)}} \]

11. Taylor expanded in s around 0

    \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(s \cdot \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)}\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \left(\frac{-1}{2} \cdot c\_n + \frac{1}{2} \cdot c\_p\right)\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot c\_n\right), \left(\frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \mathsf{+.f64}\left(\left(c\_n \cdot \frac{-1}{2}\right), \left(\frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \mathsf{+.f64}\left(\mathsf{*.f64}\left(c\_n, \frac{-1}{2}\right), \left(\frac{1}{2} \cdot c\_p\right)\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \mathsf{+.f64}\left(\mathsf{*.f64}\left(c\_n, \frac{-1}{2}\right), \left(c\_p \cdot \frac{1}{2}\right)\right)\right)\right) \]

    6. *-lowering-*.f64 98.7%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \mathsf{+.f64}\left(\mathsf{*.f64}\left(c\_n, \frac{-1}{2}\right), \mathsf{*.f64}\left(c\_p, \frac{1}{2}\right)\right)\right)\right) \]

13. Simplified 98.7%

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

Alternative 2: 95.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -1 \cdot 10^{+19}:\\ \;\;\;\;{\left(\frac{1}{2 - s}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -1e+19) (pow (/ 1.0 (- 2.0 s)) c_p) 1.0))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -1e+19) {
		tmp = pow((1.0 / (2.0 - s)), c_p);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(c_p, c_n, t, s)
    real(8), intent (in) :: c_p
    real(8), intent (in) :: c_n
    real(8), intent (in) :: t
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= (-1d+19)) then
        tmp = (1.0d0 / (2.0d0 - s)) ** c_p
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -1e+19) {
		tmp = Math.pow((1.0 / (2.0 - s)), c_p);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(c_p, c_n, t, s):
	tmp = 0
	if s <= -1e+19:
		tmp = math.pow((1.0 / (2.0 - s)), c_p)
	else:
		tmp = 1.0
	return tmp

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -1e+19)
		tmp = Float64(1.0 / Float64(2.0 - s)) ^ c_p;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c_p, c_n, t, s)
	tmp = 0.0;
	if (s <= -1e+19)
		tmp = (1.0 / (2.0 - s)) ^ c_p;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -1e+19], N[Power[N[(1.0 / N[(2.0 - s), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if s < -1e19

   1. Initial program 57.1%

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

   3. Taylor expanded in c_p around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(s\right)\right)\right)\right)\right), c\_n\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(t\right)\right)\right)\right)\right), c\_n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 100.0%

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

   6. Taylor expanded in c_n around 0

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

      1. neg-mul-1 N/A

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

      2. neg-mul-1 N/A

         \[\leadsto {\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p} \]

      3. rec-exp N/A

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

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\left(\frac{1}{1 + \frac{1}{e^{s}}}\right), \color{blue}{c\_p}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \frac{1}{e^{s}}\right)\right), c\_p\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right) \]

      7. rec-exp N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{-1 \cdot s}\right)\right)\right), c\_p\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(-1 \cdot s\right)\right)\right)\right), c\_p\right) \]

      10. neg-mul-1 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), c\_p\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - s\right)\right)\right)\right), c\_p\right) \]

      12. --lowering--.f64 100.0%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, s\right)\right)\right)\right), c\_p\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{0 - s}}\right)}^{c\_p}} \]

   9. Taylor expanded in s around 0

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(2 + -1 \cdot s\right)}\right), c\_p\right) \]
   10. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(\mathsf{neg}\left(s\right)\right)\right)\right), c\_p\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(2 - s\right)\right), c\_p\right) \]

       3. --lowering--.f64 72.4%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(2, s\right)\right), c\_p\right) \]

   11. Simplified 72.4%

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

   if -1e19 < s

   1. Initial program 92.8%

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

   3. Taylor expanded in c_n around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}\right), \color{blue}{\left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)}\right) \]

      2. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_p\right), \left({\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right), c\_p\right), \left({\left(\frac{\color{blue}{1}}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_p}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\mathsf{neg}\left(t\right)}\right)\right), c\_p\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(t\right)}\right)\right)\right), c\_p\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), c\_p\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - t\right)\right)\right)\right), c\_p\right)\right) \]

      13. --lowering--.f64 94.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, t\right)\right)\right)\right), c\_p\right)\right) \]

   5. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + \frac{1}{e^{s}}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{0 - t}}\right)}^{c\_p}}} \]

   6. Taylor expanded in c_p around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 97.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.6% accurate, 7.8× speedup

Math

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

FPCore

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

C

double code(double c_p, double c_n, double t, double s) {
	return exp((s * (s * (c_n * -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 * (s * (c_n * (-0.125d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

4. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{e^{c\_p \cdot \left(\left(0 - \mathsf{log1p}\left(\frac{1}{e^{s}}\right)\right) - \left(0 - \mathsf{log1p}\left(e^{0 - t}\right)\right)\right) + c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{s}}}\right) - \mathsf{log1p}\left(\frac{1}{-1 + \frac{-1}{e^{t}}}\right)\right)}} \]

5. Taylor expanded in s around 0

   \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c\_p, \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, t\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(c\_n, \color{blue}{\left(\left(\log \frac{1}{2} + s \cdot \left(s \cdot \left(\frac{1}{192} \cdot {s}^{2} - \frac{1}{8}\right) - \frac{1}{2}\right)\right) - \log \left(1 - \frac{1}{1 + \frac{1}{e^{t}}}\right)\right)}\right)\right)\right) \]

7. Simplified 93.0%

   \[\leadsto e^{c\_p \cdot \left(\left(0 - \mathsf{log1p}\left(\frac{1}{e^{s}}\right)\right) - \left(0 - \mathsf{log1p}\left(e^{0 - t}\right)\right)\right) + c\_n \cdot \color{blue}{\left(s \cdot \left(s \cdot \left(\left(s \cdot s\right) \cdot 0.005208333333333333 + -0.125\right) + -0.5\right) + \left(\log 0.5 - \mathsf{log1p}\left(\frac{-1}{1 + e^{0 - t}}\right)\right)\right)}} \]

8. Taylor expanded in s around inf

   \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left({s}^{4} \cdot \left(\frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}} + \frac{1}{192} \cdot c\_n\right)\right)}\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({s}^{4}\right), \left(\frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}} + \frac{1}{192} \cdot c\_n\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({s}^{\left(2 \cdot 2\right)}\right), \left(\frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}} + \frac{1}{192} \cdot c\_n\right)\right)\right) \]

   3. pow-sqr N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({s}^{2} \cdot {s}^{2}\right), \left(\frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}} + \frac{1}{192} \cdot c\_n\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({s}^{2}\right), \left({s}^{2}\right)\right), \left(\frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}} + \frac{1}{192} \cdot c\_n\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(s \cdot s\right), \left({s}^{2}\right)\right), \left(\frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}} + \frac{1}{192} \cdot c\_n\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \left({s}^{2}\right)\right), \left(\frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}} + \frac{1}{192} \cdot c\_n\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \left(s \cdot s\right)\right), \left(\frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}} + \frac{1}{192} \cdot c\_n\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \left(\frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}} + \frac{1}{192} \cdot c\_n\right)\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \left(\frac{1}{192} \cdot c\_n + \frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}}\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \left(c\_n \cdot \frac{1}{192} + \frac{-1}{8} \cdot \frac{c\_n}{{s}^{2}}\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \left(c\_n \cdot \frac{1}{192} + \frac{\frac{-1}{8} \cdot c\_n}{{s}^{2}}\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \left(c\_n \cdot \frac{1}{192} + \frac{c\_n \cdot \frac{-1}{8}}{{s}^{2}}\right)\right)\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \left(c\_n \cdot \frac{1}{192} + c\_n \cdot \frac{\frac{-1}{8}}{{s}^{2}}\right)\right)\right) \]

   14. distribute-lft-out N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \left(c\_n \cdot \left(\frac{1}{192} + \frac{\frac{-1}{8}}{{s}^{2}}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \mathsf{*.f64}\left(c\_n, \left(\frac{1}{192} + \frac{\frac{-1}{8}}{{s}^{2}}\right)\right)\right)\right) \]

   16. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \mathsf{*.f64}\left(c\_n, \mathsf{+.f64}\left(\frac{1}{192}, \left(\frac{\frac{-1}{8}}{{s}^{2}}\right)\right)\right)\right)\right) \]

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \mathsf{*.f64}\left(c\_n, \mathsf{+.f64}\left(\frac{1}{192}, \mathsf{/.f64}\left(\frac{-1}{8}, \left({s}^{2}\right)\right)\right)\right)\right)\right) \]

   18. unpow2 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \mathsf{*.f64}\left(c\_n, \mathsf{+.f64}\left(\frac{1}{192}, \mathsf{/.f64}\left(\frac{-1}{8}, \left(s \cdot s\right)\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f64 41.3%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(s, s\right), \mathsf{*.f64}\left(s, s\right)\right), \mathsf{*.f64}\left(c\_n, \mathsf{+.f64}\left(\frac{1}{192}, \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(s, s\right)\right)\right)\right)\right)\right) \]

10. Simplified 41.3%

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

11. Taylor expanded in s around 0

    \[\leadsto \mathsf{exp.f64}\left(\color{blue}{\left(\frac{-1}{8} \cdot \left(c\_n \cdot {s}^{2}\right)\right)}\right) \]
12. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(\left(\frac{-1}{8} \cdot c\_n\right) \cdot {s}^{2}\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(\left(\frac{-1}{8} \cdot c\_n\right) \cdot \left(s \cdot s\right)\right)\right) \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(\left(\left(\frac{-1}{8} \cdot c\_n\right) \cdot s\right) \cdot s\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{exp.f64}\left(\left(s \cdot \left(\left(\frac{-1}{8} \cdot c\_n\right) \cdot s\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \left(\left(\frac{-1}{8} \cdot c\_n\right) \cdot s\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \left(s \cdot \left(\frac{-1}{8} \cdot c\_n\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \left(\frac{-1}{8} \cdot c\_n\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \left(c\_n \cdot \frac{-1}{8}\right)\right)\right)\right) \]

    9. *-lowering-*.f64 98.3%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(s, \mathsf{*.f64}\left(s, \mathsf{*.f64}\left(c\_n, \frac{-1}{8}\right)\right)\right)\right) \]

13. Simplified 98.3%

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

Alternative 4: 94.0% accurate, 835.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

3. Taylor expanded in c_n around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right)}^{c\_p}\right), \color{blue}{\left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)}\right) \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(s\right)}}\right), c\_p\right), \left({\color{blue}{\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}}^{c\_p}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\mathsf{neg}\left(s\right)}\right)\right), c\_p\right), \left({\left(\frac{\color{blue}{1}}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(s\right)}\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{e^{s}}\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{s}\right)\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   7. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \left({\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right)}^{c\_p}\right)\right) \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\left(\frac{1}{1 + e^{\mathsf{neg}\left(t\right)}}\right), \color{blue}{c\_p}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \left(1 + e^{\mathsf{neg}\left(t\right)}\right)\right), c\_p\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(e^{\mathsf{neg}\left(t\right)}\right)\right)\right), c\_p\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right)\right), c\_p\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - t\right)\right)\right)\right), c\_p\right)\right) \]

   13. --lowering--.f64 93.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(s\right)\right)\right)\right), c\_p\right), \mathsf{pow.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, t\right)\right)\right)\right), c\_p\right)\right) \]

5. Simplified 93.0%

   \[\leadsto \color{blue}{\frac{{\left(\frac{1}{1 + \frac{1}{e^{s}}}\right)}^{c\_p}}{{\left(\frac{1}{1 + e^{0 - t}}\right)}^{c\_p}}} \]

6. Taylor expanded in c_p around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 94.5%

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

Developer Target 1: 96.2% 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 2024191 
(FPCore (c_p c_n t s)
  :name "Harley's example"
  :precision binary64
  :pre (and (< 0.0 c_p) (< 0.0 c_n))

  :alt
  (! :herbie-platform default (* (pow (/ (+ 1 (exp (- t))) (+ 1 (exp (- s)))) c_p) (pow (/ (+ 1 (exp t)) (+ 1 (exp s))) c_n)))

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