Harley's example

Percentage Accurate: 90.6% → 98.6%

Time: 52.8s

Alternatives: 8

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 3.6220849208399805e-165
  2. c_n = 2.9500981734005963e-255
  3. t = 1.0505279143868682e+139
  4. s = 9.04290039963357e-261

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: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s}\\ \mathbf{if}\;-s \leq 200000000:\\ \;\;\;\;e^{\left(\log \left(1 - e^{-\mathsf{log1p}\left(t\_1\right)}\right) - \log \left(1 - e^{-\mathsf{log1p}\left(e^{-t}\right)}\right)\right) \cdot c\_n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(t\_1 + 1\right) \cdot \mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- s))))
   (if (<= (- s) 200000000.0)
     (exp
      (*
       (-
        (log (- 1.0 (exp (- (log1p t_1)))))
        (log (- 1.0 (exp (- (log1p (exp (- t))))))))
       c_n))
     (pow (pow (* (+ t_1 1.0) (fma 0.25 t 0.5)) c_p) -1.0))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-s);
	double tmp;
	if (-s <= 200000000.0) {
		tmp = exp(((log((1.0 - exp(-log1p(t_1)))) - log((1.0 - exp(-log1p(exp(-t)))))) * c_n));
	} else {
		tmp = pow(pow(((t_1 + 1.0) * fma(0.25, t, 0.5)), c_p), -1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-s))
	tmp = 0.0
	if (Float64(-s) <= 200000000.0)
		tmp = exp(Float64(Float64(log(Float64(1.0 - exp(Float64(-log1p(t_1))))) - log(Float64(1.0 - exp(Float64(-log1p(exp(Float64(-t)))))))) * c_n));
	else
		tmp = (Float64(Float64(t_1 + 1.0) * fma(0.25, t, 0.5)) ^ c_p) ^ -1.0;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[(-s)], $MachinePrecision]}, If[LessEqual[(-s), 200000000.0], N[Exp[N[(N[(N[Log[N[(1.0 - N[Exp[(-N[Log[1 + t$95$1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(1.0 - N[Exp[(-N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c$95$n), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[N[(N[(t$95$1 + 1.0), $MachinePrecision] * N[(0.25 * t + 0.5), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 2e8

   1. Initial program 90.9%

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in c_p around 0

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

   7. Applied rewrites 99.6%

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

   if 2e8 < (neg.f64 s)

   1. Initial program 50.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. 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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{2} + \frac{1}{4} \cdot t\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.7%

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.7%

       \[\leadsto \color{blue}{\frac{{\left(e^{-s} + 1\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\color{blue}{{\left(\left(e^{-s} + 1\right) \cdot \mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 200000000:\\ \;\;\;\;e^{\left(\log \left(1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}\right) - \log \left(1 - e^{-\mathsf{log1p}\left(e^{-t}\right)}\right)\right) \cdot c\_n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\left(e^{-s} + 1\right) \cdot \mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -740000000:\\ \;\;\;\;{\left({\left(\left(e^{-s} + 1\right) \cdot \mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\mathsf{fma}\left(s, -0.5, \log 0.5\right) - \log 0.5\right) \cdot c\_n}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -740000000.0)
   (pow (pow (* (+ (exp (- s)) 1.0) (fma 0.25 t 0.5)) c_p) -1.0)
   (exp (* (- (fma s -0.5 (log 0.5)) (log 0.5)) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -740000000.0) {
		tmp = pow(pow(((exp(-s) + 1.0) * fma(0.25, t, 0.5)), c_p), -1.0);
	} else {
		tmp = exp(((fma(s, -0.5, log(0.5)) - log(0.5)) * c_n));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -740000000.0)
		tmp = (Float64(Float64(exp(Float64(-s)) + 1.0) * fma(0.25, t, 0.5)) ^ c_p) ^ -1.0;
	else
		tmp = exp(Float64(Float64(fma(s, -0.5, log(0.5)) - log(0.5)) * c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -740000000.0], N[Power[N[Power[N[(N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.25 * t + 0.5), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], -1.0], $MachinePrecision], N[Exp[N[(N[(N[(s * -0.5 + N[Log[0.5], $MachinePrecision]), $MachinePrecision] - N[Log[0.5], $MachinePrecision]), $MachinePrecision] * c$95$n), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -7.4e8

   1. Initial program 50.0%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. 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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{2} + \frac{1}{4} \cdot t\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.7%

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.7%

       \[\leadsto \color{blue}{\frac{{\left(e^{-s} + 1\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\color{blue}{{\left(\left(e^{-s} + 1\right) \cdot \mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}} \]

   if -7.4e8 < s

   1. Initial program 90.9%

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in c_p around 0

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in t around 0

      \[\leadsto e^{\left(\log \left(1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}\right) - \log \left(1 - e^{\mathsf{neg}\left(\log 2\right)}\right)\right) \cdot c\_n} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.7%

       \[\leadsto e^{\left(\log \left(1 - e^{-\mathsf{log1p}\left(e^{-s}\right)}\right) - \log 0.5\right) \cdot c\_n} \]

   11. Taylor expanded in s around 0

       \[\leadsto e^{\left(\left(\log \left(1 - e^{\mathsf{neg}\left(\log 2\right)}\right) + \frac{-1}{2} \cdot \frac{s \cdot e^{\mathsf{neg}\left(\log 2\right)}}{1 - e^{\mathsf{neg}\left(\log 2\right)}}\right) - \log \frac{1}{2}\right) \cdot c\_n} \]
   12. Step-by-step derivation

   13. Applied rewrites 98.9%

       \[\leadsto e^{\left(\mathsf{fma}\left(s \cdot 1, -0.5, \log 0.5\right) - \log 0.5\right) \cdot c\_n} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq -740000000:\\ \;\;\;\;{\left({\left(\left(e^{-s} + 1\right) \cdot \mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\mathsf{fma}\left(s, -0.5, \log 0.5\right) - \log 0.5\right) \cdot c\_n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -4 \cdot 10^{-30}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right) \cdot s - 1, s, 2\right)\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -4e-30)
   (/
    (pow (fma (- (* (fma -0.16666666666666666 s 0.5) s) 1.0) s 2.0) (- c_p))
    (pow (fma 0.25 t 0.5) c_p))
   (/ (pow 0.5 c_n) (pow (fma -0.25 t 0.5) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -4e-30) {
		tmp = pow(fma(((fma(-0.16666666666666666, s, 0.5) * s) - 1.0), s, 2.0), -c_p) / pow(fma(0.25, t, 0.5), c_p);
	} else {
		tmp = pow(0.5, c_n) / pow(fma(-0.25, t, 0.5), c_n);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -4e-30)
		tmp = Float64((fma(Float64(Float64(fma(-0.16666666666666666, s, 0.5) * s) - 1.0), s, 2.0) ^ Float64(-c_p)) / (fma(0.25, t, 0.5) ^ c_p));
	else
		tmp = Float64((0.5 ^ c_n) / (fma(-0.25, t, 0.5) ^ c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -4e-30], N[(N[Power[N[(N[(N[(N[(-0.16666666666666666 * s + 0.5), $MachinePrecision] * s), $MachinePrecision] - 1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[N[(0.25 * t + 0.5), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(-0.25 * t + 0.5), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -4e-30

   1. Initial program 68.4%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. 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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{2} + \frac{1}{4} \cdot t\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.2%

       \[\leadsto \color{blue}{\frac{{\left(e^{-s} + 1\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}} \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{{\left(2 + s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right)\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4}}, t, \frac{1}{2}\right)\right)}^{c\_p}} \]
   12. Step-by-step derivation

   13. Applied rewrites 87.0%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, s, 0.5\right) \cdot s - 1, s, 2\right)\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(\color{blue}{0.25}, t, 0.5\right)\right)}^{c\_p}} \]

   if -4e-30 < s

   1. Initial program 92.6%

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

   3. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-pow.f64 N/A

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\color{blue}{\left(1 - \frac{1}{e^{-t} + 1}\right)}}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in t around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.3%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 2 \cdot 10^{-107}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 2e-107)
   (/ (pow 0.5 c_n) (pow (fma -0.25 t 0.5) c_n))
   (/ (pow (fma (- (* 0.5 s) 1.0) s 2.0) (- c_p)) (pow (fma 0.25 t 0.5) c_p))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 2e-107) {
		tmp = pow(0.5, c_n) / pow(fma(-0.25, t, 0.5), c_n);
	} else {
		tmp = pow(fma(((0.5 * s) - 1.0), s, 2.0), -c_p) / pow(fma(0.25, t, 0.5), c_p);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 2e-107)
		tmp = Float64((0.5 ^ c_n) / (fma(-0.25, t, 0.5) ^ c_n));
	else
		tmp = Float64((fma(Float64(Float64(0.5 * s) - 1.0), s, 2.0) ^ Float64(-c_p)) / (fma(0.25, t, 0.5) ^ c_p));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 2e-107], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(-0.25 * t + 0.5), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(0.5 * s), $MachinePrecision] - 1.0), $MachinePrecision] * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[N[(0.25 * t + 0.5), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 2e-107

   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. Add Preprocessing

   3. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-pow.f64 N/A

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\color{blue}{\left(1 - \frac{1}{e^{-t} + 1}\right)}}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in t around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.0%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}} \]

   if 2e-107 < (neg.f64 s)

   1. Initial program 84.8%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{2} + \frac{1}{4} \cdot t\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}} \]
   9. Step-by-step derivation

   10. Applied rewrites 87.1%

       \[\leadsto \color{blue}{\frac{{\left(e^{-s} + 1\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}} \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{{\left(2 + s \cdot \left(\frac{1}{2} \cdot s - 1\right)\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4}}, t, \frac{1}{2}\right)\right)}^{c\_p}} \]
   12. Step-by-step derivation

   13. Applied rewrites 93.7%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(0.5 \cdot s - 1, s, 2\right)\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(\color{blue}{0.25}, t, 0.5\right)\right)}^{c\_p}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -4 \cdot 10^{-30}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(-1, s, 2\right)\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -4e-30)
   (/ (pow (fma -1.0 s 2.0) (- c_p)) (pow (fma 0.25 t 0.5) c_p))
   (/ (pow 0.5 c_n) (pow (fma -0.25 t 0.5) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -4e-30) {
		tmp = pow(fma(-1.0, s, 2.0), -c_p) / pow(fma(0.25, t, 0.5), c_p);
	} else {
		tmp = pow(0.5, c_n) / pow(fma(-0.25, t, 0.5), c_n);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -4e-30)
		tmp = Float64((fma(-1.0, s, 2.0) ^ Float64(-c_p)) / (fma(0.25, t, 0.5) ^ c_p));
	else
		tmp = Float64((0.5 ^ c_n) / (fma(-0.25, t, 0.5) ^ c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -4e-30], N[(N[Power[N[(-1.0 * s + 2.0), $MachinePrecision], (-c$95$p)], $MachinePrecision] / N[Power[N[(0.25 * t + 0.5), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(-0.25 * t + 0.5), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -4e-30

   1. Initial program 68.4%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. 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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{2} + \frac{1}{4} \cdot t\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.2%

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.2%

       \[\leadsto \color{blue}{\frac{{\left(e^{-s} + 1\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}} \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{{\left(2 + -1 \cdot s\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4}}, t, \frac{1}{2}\right)\right)}^{c\_p}} \]
   12. Step-by-step derivation

   13. Applied rewrites 87.0%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(-1, s, 2\right)\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(\color{blue}{0.25}, t, 0.5\right)\right)}^{c\_p}} \]

   if -4e-30 < s

   1. Initial program 92.6%

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

   3. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-pow.f64 N/A

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\color{blue}{\left(1 - \frac{1}{e^{-t} + 1}\right)}}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in t around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.3%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-t \leq -5 \cdot 10^{-140}:\\ \;\;\;\;\frac{{2}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- t) -5e-140)
   (/ (pow 2.0 (- c_p)) (pow (fma 0.25 t 0.5) c_p))
   (/ (pow 0.5 c_n) (pow (fma -0.25 t 0.5) c_n))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-t <= -5e-140) {
		tmp = pow(2.0, -c_p) / pow(fma(0.25, t, 0.5), c_p);
	} else {
		tmp = pow(0.5, c_n) / pow(fma(-0.25, t, 0.5), c_n);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-t) <= -5e-140)
		tmp = Float64((2.0 ^ Float64(-c_p)) / (fma(0.25, t, 0.5) ^ c_p));
	else
		tmp = Float64((0.5 ^ c_n) / (fma(-0.25, t, 0.5) ^ c_n));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-t), -5e-140], N[(N[Power[2.0, (-c$95$p)], $MachinePrecision] / N[Power[N[(0.25 * t + 0.5), $MachinePrecision], c$95$p], $MachinePrecision]), $MachinePrecision], N[(N[Power[0.5, c$95$n], $MachinePrecision] / N[Power[N[(-0.25 * t + 0.5), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 t) < -5.00000000000000015e-140

   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. 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. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 95.0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\frac{1}{2} + \frac{1}{4} \cdot t\right)}^{c\_p}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.0%

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.0%

       \[\leadsto \color{blue}{\frac{{\left(e^{-s} + 1\right)}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(0.25, t, 0.5\right)\right)}^{c\_p}}} \]

   11. Taylor expanded in s around 0

       \[\leadsto \frac{{2}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4}}, t, \frac{1}{2}\right)\right)}^{c\_p}} \]
   12. Step-by-step derivation

   13. Applied rewrites 98.3%

       \[\leadsto \frac{{2}^{\left(-c\_p\right)}}{{\left(\mathsf{fma}\left(\color{blue}{0.25}, t, 0.5\right)\right)}^{c\_p}} \]

   if -5.00000000000000015e-140 < (neg.f64 t)

   1. Initial program 89.5%

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

   3. Taylor expanded in c_p around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-pow.f64 N/A

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in s around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\color{blue}{\left(1 - \frac{1}{e^{-t} + 1}\right)}}^{c\_n}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.7%

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

   9. Taylor expanded in t around 0

      \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]
   10. Step-by-step derivation

   11. Applied rewrites 95.7%

       \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (/ (pow 0.5 c_n) (pow (fma -0.25 t 0.5) c_n)))

C

double code(double c_p, double c_n, double t, double s) {
	return pow(0.5, c_n) / pow(fma(-0.25, t, 0.5), c_n);
}

Julia

function code(c_p, c_n, t, s)
	return Float64((0.5 ^ c_n) / (fma(-0.25, t, 0.5) ^ c_n))
end

Wolfram

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

Derivation

1. Initial program 90.0%

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

3. Taylor expanded in c_p around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-pow.f64 N/A

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

5. Applied rewrites 95.8%

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

6. Taylor expanded in s around 0

   \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\color{blue}{\left(1 - \frac{1}{e^{-t} + 1}\right)}}^{c\_n}} \]
7. Step-by-step derivation

8. Applied rewrites 95.5%

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

9. Taylor expanded in t around 0

   \[\leadsto \frac{{\frac{1}{2}}^{c\_n}}{{\left(\frac{1}{2} + \frac{-1}{4} \cdot t\right)}^{c\_n}} \]
10. Step-by-step derivation

11. Applied rewrites 94.7%

    \[\leadsto \frac{{0.5}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}} \]
12. Add Preprocessing

Alternative 8: 93.9% accurate, 896.0× 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 90.0%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
2. 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. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 90.5

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

5. Applied rewrites 90.5%

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

6. Taylor expanded in c_p around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 94.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer Target 1: 96.4% accurate, 1.4× 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 2024333 
(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))))