Harley's example

Percentage Accurate: 91.3% → 96.3%

Time: 52.4s

Alternatives: 8

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 25795701.96364263
  2. c_n = 4.820078163477958e+69
  3. t = -1.1318370884348916e+138
  4. s = 2.3116558903060748e+172

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.3% 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: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{-s}\\ t_2 := e^{-t}\\ e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(t\_1\right)\right) + \mathsf{log1p}\left(t\_2\right), c\_n \cdot \left(\mathsf{log1p}\left({\left(-1 - t\_1\right)}^{-1}\right) - \mathsf{log1p}\left({\left(-1 - t\_2\right)}^{-1}\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- s))) (t_2 (exp (- t))))
   (exp
    (fma
     c_p
     (+ (- (log1p t_1)) (log1p t_2))
     (*
      c_n
      (- (log1p (pow (- -1.0 t_1) -1.0)) (log1p (pow (- -1.0 t_2) -1.0))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp(-s);
	double t_2 = exp(-t);
	return exp(fma(c_p, (-log1p(t_1) + log1p(t_2)), (c_n * (log1p(pow((-1.0 - t_1), -1.0)) - log1p(pow((-1.0 - t_2), -1.0))))));
}

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(-s))
	t_2 = exp(Float64(-t))
	return exp(fma(c_p, Float64(Float64(-log1p(t_1)) + log1p(t_2)), Float64(c_n * Float64(log1p((Float64(-1.0 - t_1) ^ -1.0)) - log1p((Float64(-1.0 - t_2) ^ -1.0))))))
end

Wolfram

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

Derivation

1. Initial program 93.0%

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

   \[\leadsto e^{\mathsf{fma}\left(c\_p, \left(-\mathsf{log1p}\left(e^{-s}\right)\right) + \mathsf{log1p}\left(e^{-t}\right), c\_n \cdot \left(\mathsf{log1p}\left({\left(-1 - e^{-s}\right)}^{-1}\right) - \mathsf{log1p}\left({\left(-1 - e^{-t}\right)}^{-1}\right)\right)\right)} \]
6. Add Preprocessing

Alternative 2: 95.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\\ t_2 := e^{-t}\\ t_3 := {\left(1 + t\_2\right)}^{-1}\\ t_4 := {\left(1 + e^{-s}\right)}^{-1}\\ \mathbf{if}\;\frac{{t\_4}^{c\_p} \cdot {\left(1 - t\_4\right)}^{c\_n}}{{t\_3}^{c\_p} \cdot {\left(1 - t\_3\right)}^{c\_n}} \leq 1:\\ \;\;\;\;\frac{{\left(t\_1 \cdot t\_1\right)}^{\left(\left(-c\_p\right) \cdot 0.5\right)}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{2}^{\left(-c\_p\right)}}{\mathsf{fma}\left(-\mathsf{log1p}\left(t\_2\right), c\_p, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (fma (fma 0.5 s -1.0) s 2.0))
        (t_2 (exp (- t)))
        (t_3 (pow (+ 1.0 t_2) -1.0))
        (t_4 (pow (+ 1.0 (exp (- s))) -1.0)))
   (if (<=
        (/
         (* (pow t_4 c_p) (pow (- 1.0 t_4) c_n))
         (* (pow t_3 c_p) (pow (- 1.0 t_3) c_n)))
        1.0)
     (/ (pow (* t_1 t_1) (* (- c_p) 0.5)) (pow 0.5 c_p))
     (/ (pow 2.0 (- c_p)) (fma (- (log1p t_2)) c_p 1.0)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = fma(fma(0.5, s, -1.0), s, 2.0);
	double t_2 = exp(-t);
	double t_3 = pow((1.0 + t_2), -1.0);
	double t_4 = pow((1.0 + exp(-s)), -1.0);
	double tmp;
	if (((pow(t_4, c_p) * pow((1.0 - t_4), c_n)) / (pow(t_3, c_p) * pow((1.0 - t_3), c_n))) <= 1.0) {
		tmp = pow((t_1 * t_1), (-c_p * 0.5)) / pow(0.5, c_p);
	} else {
		tmp = pow(2.0, -c_p) / fma(-log1p(t_2), c_p, 1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = fma(fma(0.5, s, -1.0), s, 2.0)
	t_2 = exp(Float64(-t))
	t_3 = Float64(1.0 + t_2) ^ -1.0
	t_4 = Float64(1.0 + exp(Float64(-s))) ^ -1.0
	tmp = 0.0
	if (Float64(Float64((t_4 ^ c_p) * (Float64(1.0 - t_4) ^ c_n)) / Float64((t_3 ^ c_p) * (Float64(1.0 - t_3) ^ c_n))) <= 1.0)
		tmp = Float64((Float64(t_1 * t_1) ^ Float64(Float64(-c_p) * 0.5)) / (0.5 ^ c_p));
	else
		tmp = Float64((2.0 ^ Float64(-c_p)) / fma(Float64(-log1p(t_2)), c_p, 1.0));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[(N[(0.5 * s + -1.0), $MachinePrecision] * s + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t)], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(1.0 + t$95$2), $MachinePrecision], -1.0], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(1.0 + N[Exp[(-s)], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[t$95$4, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$4), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$3, c$95$p], $MachinePrecision] * N[Power[N[(1.0 - t$95$3), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[Power[N[(t$95$1 * t$95$1), $MachinePrecision], N[((-c$95$p) * 0.5), $MachinePrecision]], $MachinePrecision] / N[Power[0.5, c$95$p], $MachinePrecision]), $MachinePrecision], N[(N[Power[2.0, (-c$95$p)], $MachinePrecision] / N[((-N[Log[1 + t$95$2], $MachinePrecision]) * c$95$p + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n))) < 1

   1. Initial program 99.2%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. 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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in s around 0

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

   11. Applied rewrites 98.5%

       \[\leadsto \frac{{\left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)}\right)}^{c\_p}}{{0.5}^{c\_p}} \]
   12. Step-by-step derivation

   13. Applied rewrites 99.3%

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

   if 1 < (/.f64 (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 s))))) c_n)) (*.f64 (pow.f64 (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t)))) c_p) (pow.f64 (-.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (neg.f64 t))))) c_n)))

   1. Initial program 5.9%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   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. 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 11.4

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

   5. Applied rewrites 11.4%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2}\right)}^{c\_p}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.3%

       \[\leadsto \frac{{\left(\frac{1}{2}\right)}^{c\_p}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.3%

       \[\leadsto \frac{{2}^{\left(-1 \cdot c\_p\right)}}{\mathsf{fma}\left(\color{blue}{-\mathsf{log1p}\left(e^{-t}\right)}, c\_p, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{\left({\left(1 + e^{-s}\right)}^{-1}\right)}^{c\_p} \cdot {\left(1 - {\left(1 + e^{-s}\right)}^{-1}\right)}^{c\_n}}{{\left({\left(1 + e^{-t}\right)}^{-1}\right)}^{c\_p} \cdot {\left(1 - {\left(1 + e^{-t}\right)}^{-1}\right)}^{c\_n}} \leq 1:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, s, -1\right), s, 2\right)\right)}^{\left(\left(-c\_p\right) \cdot 0.5\right)}}{{0.5}^{c\_p}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{2}^{\left(-c\_p\right)}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(e^{-s} + 1\right)}^{-1}\\ \mathbf{if}\;-s \leq 200:\\ \;\;\;\;\frac{{\left(1 - t\_1\right)}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_1}^{c\_p}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (pow (+ (exp (- s)) 1.0) -1.0)))
   (if (<= (- s) 200.0)
     (/ (pow (- 1.0 t_1) c_n) (pow (fma -0.25 t 0.5) c_n))
     (/ (pow t_1 c_p) (fma (- (log1p (exp (- t)))) c_p 1.0)))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = pow((exp(-s) + 1.0), -1.0);
	double tmp;
	if (-s <= 200.0) {
		tmp = pow((1.0 - t_1), c_n) / pow(fma(-0.25, t, 0.5), c_n);
	} else {
		tmp = pow(t_1, c_p) / fma(-log1p(exp(-t)), c_p, 1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = Float64(exp(Float64(-s)) + 1.0) ^ -1.0
	tmp = 0.0
	if (Float64(-s) <= 200.0)
		tmp = Float64((Float64(1.0 - t_1) ^ c_n) / (fma(-0.25, t, 0.5) ^ c_n));
	else
		tmp = Float64((t_1 ^ c_p) / fma(Float64(-log1p(exp(Float64(-t)))), c_p, 1.0));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Power[N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[(-s), 200.0], N[(N[Power[N[(1.0 - t$95$1), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[N[(-0.25 * t + 0.5), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$1, c$95$p], $MachinePrecision] / N[((-N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]) * c$95$p + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 200

   1. Initial program 94.8%

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

   3. Taylor expanded in c_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 t around 0

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

   8. Applied rewrites 98.0%

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

   if 200 < (neg.f64 s)

   1. Initial program 16.7%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. 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 16.7

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

   5. Applied rewrites 16.7%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{{\left(\frac{1}{e^{-s} + 1}\right)}^{c\_p}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), \color{blue}{c\_p}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 200:\\ \;\;\;\;\frac{{\left(1 - {\left(e^{-s} + 1\right)}^{-1}\right)}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(e^{-s} + 1\right)}^{-1}\right)}^{c\_p}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 200:\\ \;\;\;\;\frac{{\left(1 - {\left(e^{-s} + 1\right)}^{-1}\right)}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{2}^{\left(-c\_p\right)}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 200.0)
   (/
    (pow (- 1.0 (pow (+ (exp (- s)) 1.0) -1.0)) c_n)
    (pow (fma -0.25 t 0.5) c_n))
   (/ (pow 2.0 (- c_p)) (fma (- (log1p (exp (- t)))) c_p 1.0))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 200.0) {
		tmp = pow((1.0 - pow((exp(-s) + 1.0), -1.0)), c_n) / pow(fma(-0.25, t, 0.5), c_n);
	} else {
		tmp = pow(2.0, -c_p) / fma(-log1p(exp(-t)), c_p, 1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 200.0)
		tmp = Float64((Float64(1.0 - (Float64(exp(Float64(-s)) + 1.0) ^ -1.0)) ^ c_n) / (fma(-0.25, t, 0.5) ^ c_n));
	else
		tmp = Float64((2.0 ^ Float64(-c_p)) / fma(Float64(-log1p(exp(Float64(-t)))), c_p, 1.0));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 200.0], N[(N[Power[N[(1.0 - N[Power[N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[N[(-0.25 * t + 0.5), $MachinePrecision], c$95$n], $MachinePrecision]), $MachinePrecision], N[(N[Power[2.0, (-c$95$p)], $MachinePrecision] / N[((-N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]) * c$95$p + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 200

   1. Initial program 94.8%

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

   3. Taylor expanded in c_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 t around 0

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

   8. Applied rewrites 98.0%

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

   if 200 < (neg.f64 s)

   1. Initial program 16.7%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. 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 16.7

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

   5. Applied rewrites 16.7%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2}\right)}^{c\_p}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 83.9%

       \[\leadsto \frac{{\left(\frac{1}{2}\right)}^{c\_p}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 83.9%

       \[\leadsto \frac{{2}^{\left(-1 \cdot c\_p\right)}}{\mathsf{fma}\left(\color{blue}{-\mathsf{log1p}\left(e^{-t}\right)}, c\_p, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 200:\\ \;\;\;\;\frac{{\left(1 - {\left(e^{-s} + 1\right)}^{-1}\right)}^{c\_n}}{{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{2}^{\left(-c\_p\right)}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-s \leq 200:\\ \;\;\;\;\frac{{\left(1 - {\left(e^{-s} + 1\right)}^{-1}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{2}^{\left(-c\_p\right)}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= (- s) 200.0)
   (/ (pow (- 1.0 (pow (+ (exp (- s)) 1.0) -1.0)) c_n) (pow 0.5 c_n))
   (/ (pow 2.0 (- c_p)) (fma (- (log1p (exp (- t)))) c_p 1.0))))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (-s <= 200.0) {
		tmp = pow((1.0 - pow((exp(-s) + 1.0), -1.0)), c_n) / pow(0.5, c_n);
	} else {
		tmp = pow(2.0, -c_p) / fma(-log1p(exp(-t)), c_p, 1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (Float64(-s) <= 200.0)
		tmp = Float64((Float64(1.0 - (Float64(exp(Float64(-s)) + 1.0) ^ -1.0)) ^ c_n) / (0.5 ^ c_n));
	else
		tmp = Float64((2.0 ^ Float64(-c_p)) / fma(Float64(-log1p(exp(Float64(-t)))), c_p, 1.0));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[(-s), 200.0], N[(N[Power[N[(1.0 - N[Power[N[(N[Exp[(-s)], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], c$95$n], $MachinePrecision] / N[Power[0.5, c$95$n], $MachinePrecision]), $MachinePrecision], N[(N[Power[2.0, (-c$95$p)], $MachinePrecision] / N[((-N[Log[1 + N[Exp[(-t)], $MachinePrecision]], $MachinePrecision]) * c$95$p + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 s) < 200

   1. Initial program 94.8%

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

   3. Taylor expanded in c_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 t around 0

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

   8. Applied rewrites 97.2%

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

   if 200 < (neg.f64 s)

   1. Initial program 16.7%

      \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
   2. 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 16.7

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

   5. Applied rewrites 16.7%

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

   6. Taylor expanded in c_p around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in s around 0

      \[\leadsto \frac{{\left(\frac{1}{2}\right)}^{c\_p}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 83.9%

       \[\leadsto \frac{{\left(\frac{1}{2}\right)}^{c\_p}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 83.9%

       \[\leadsto \frac{{2}^{\left(-1 \cdot c\_p\right)}}{\mathsf{fma}\left(\color{blue}{-\mathsf{log1p}\left(e^{-t}\right)}, c\_p, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-s \leq 200:\\ \;\;\;\;\frac{{\left(1 - {\left(e^{-s} + 1\right)}^{-1}\right)}^{c\_n}}{{0.5}^{c\_n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{2}^{\left(-c\_p\right)}}{\mathsf{fma}\left(-\mathsf{log1p}\left(e^{-t}\right), c\_p, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.5% 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 93.0%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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 t around 0

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

8. Applied rewrites 95.8%

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

9. Taylor expanded in s around 0

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

11. Applied rewrites 94.8%

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

Alternative 7: 93.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.0%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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 t around 0

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

8. Applied rewrites 95.8%

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

9. Taylor expanded in s around 0

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

11. Applied rewrites 94.8%

    \[\leadsto \frac{{0.5}^{c\_n}}{{\color{blue}{\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}}^{c\_n}} \]
12. Step-by-step derivation

13. Applied rewrites 94.8%

    \[\leadsto {\left(\mathsf{fma}\left(-0.25, t, 0.5\right)\right)}^{\left(-c\_n\right)} \cdot \color{blue}{{0.5}^{c\_n}} \]
14. Add Preprocessing

Alternative 8: 94.6% 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 93.0%

   \[\frac{{\left(\frac{1}{1 + e^{-s}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-s}}\right)}^{c\_n}}{{\left(\frac{1}{1 + e^{-t}}\right)}^{c\_p} \cdot {\left(1 - \frac{1}{1 + e^{-t}}\right)}^{c\_n}} \]
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 92.6

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

5. Applied rewrites 92.6%

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

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: 96.8% 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 2024326 
(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))))