Harley's example

Percentage Accurate: 91.3% → 97.7%

Time: 53.6s

Alternatives: 8

Speedup: 896.0×

• could not determine a ground truth

  1. c_p = 4.4209138088390083e-184
  2. c_n = 2.782649528213914e-288
  3. t = 1.691144237545281e+149
  4. s = 1.3209486902291682e-40

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: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{0 - t}\\ t_2 := e^{0 - s}\\ \mathbf{if}\;c\_p \leq 4 \cdot 10^{-77}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 0.5, -1\right), 2\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(t\_1\right) - \mathsf{log1p}\left(t\_2\right), c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 - t\_2}\right) - \mathsf{log1p}\left(\frac{1}{-1 - t\_1}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (let* ((t_1 (exp (- 0.0 t))) (t_2 (exp (- 0.0 s))))
   (if (<= c_p 4e-77)
     (pow (/ 1.0 (fma s (fma s 0.5 -1.0) 2.0)) c_p)
     (exp
      (fma
       c_p
       (- (log1p t_1) (log1p t_2))
       (*
        c_n
        (- (log1p (/ 1.0 (- -1.0 t_2))) (log1p (/ 1.0 (- -1.0 t_1))))))))))

C

double code(double c_p, double c_n, double t, double s) {
	double t_1 = exp((0.0 - t));
	double t_2 = exp((0.0 - s));
	double tmp;
	if (c_p <= 4e-77) {
		tmp = pow((1.0 / fma(s, fma(s, 0.5, -1.0), 2.0)), c_p);
	} else {
		tmp = exp(fma(c_p, (log1p(t_1) - log1p(t_2)), (c_n * (log1p((1.0 / (-1.0 - t_2))) - log1p((1.0 / (-1.0 - t_1)))))));
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	t_1 = exp(Float64(0.0 - t))
	t_2 = exp(Float64(0.0 - s))
	tmp = 0.0
	if (c_p <= 4e-77)
		tmp = Float64(1.0 / fma(s, fma(s, 0.5, -1.0), 2.0)) ^ c_p;
	else
		tmp = exp(fma(c_p, Float64(log1p(t_1) - log1p(t_2)), Float64(c_n * Float64(log1p(Float64(1.0 / Float64(-1.0 - t_2))) - log1p(Float64(1.0 / Float64(-1.0 - t_1)))))));
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := Block[{t$95$1 = N[Exp[N[(0.0 - t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(0.0 - s), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$p, 4e-77], N[Power[N[(1.0 / N[(s * N[(s * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], c$95$p], $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[(1.0 / N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c_p < 3.9999999999999997e-77

   1. Initial program 92.0%

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

   3. Taylor expanded in c_p around 0

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

   5. Simplified 93.8%

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

   6. Taylor expanded in c_n around 0

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

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

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

      2. neg-mul-1 N/A

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

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

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

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

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

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

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

      6. neg-mul-1 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 94.4

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

   8. Simplified 94.4%

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

   9. Taylor expanded in s around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. accelerator-lowering-fma.f64 98.3

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

   11. Simplified 98.3%

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

   if 3.9999999999999997e-77 < c_p

   1. Initial program 84.3%

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

   4. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(c\_p, \left(0 - \mathsf{log1p}\left(e^{0 - s}\right)\right) - \left(0 - \mathsf{log1p}\left(e^{0 - t}\right)\right), c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 - e^{0 - s}}\right) - \mathsf{log1p}\left(\frac{1}{-1 - e^{0 - t}}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c\_p \leq 4 \cdot 10^{-77}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 0.5, -1\right), 2\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(c\_p, \mathsf{log1p}\left(e^{0 - t}\right) - \mathsf{log1p}\left(e^{0 - s}\right), c\_n \cdot \left(\mathsf{log1p}\left(\frac{1}{-1 - e^{0 - s}}\right) - \mathsf{log1p}\left(\frac{1}{-1 - e^{0 - t}}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c\_p \leq 5 \cdot 10^{-20}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 0.5, -1\right), 2\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(1 + e^{0 - t}\right) \cdot \frac{1}{1 + e^{0 - s}}\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= c_p 5e-20)
   (pow (/ 1.0 (fma s (fma s 0.5 -1.0) 2.0)) c_p)
   (pow (* (+ 1.0 (exp (- 0.0 t))) (/ 1.0 (+ 1.0 (exp (- 0.0 s))))) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (c_p <= 5e-20) {
		tmp = pow((1.0 / fma(s, fma(s, 0.5, -1.0), 2.0)), c_p);
	} else {
		tmp = pow(((1.0 + exp((0.0 - t))) * (1.0 / (1.0 + exp((0.0 - s))))), c_p);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (c_p <= 5e-20)
		tmp = Float64(1.0 / fma(s, fma(s, 0.5, -1.0), 2.0)) ^ c_p;
	else
		tmp = Float64(Float64(1.0 + exp(Float64(0.0 - t))) * Float64(1.0 / Float64(1.0 + exp(Float64(0.0 - s))))) ^ c_p;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[c$95$p, 5e-20], N[Power[N[(1.0 / N[(s * N[(s * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[Power[N[(N[(1.0 + N[Exp[N[(0.0 - t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[Exp[N[(0.0 - s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c_p < 4.9999999999999999e-20

   1. Initial program 91.9%

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

   3. Taylor expanded in c_p around 0

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

   5. Simplified 94.0%

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

   6. Taylor expanded in c_n around 0

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

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

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

      2. neg-mul-1 N/A

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

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

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

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

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

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

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

      6. neg-mul-1 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 94.9

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

   8. Simplified 94.9%

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

   9. Taylor expanded in s around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. accelerator-lowering-fma.f64 97.9

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

   11. Simplified 97.9%

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

   if 4.9999999999999999e-20 < c_p

   1. Initial program 67.9%

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

   3. Taylor expanded in c_n around 0

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

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

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

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

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

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

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

      6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. --lowering--.f64 68.0

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

   5. Simplified 68.0%

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

   7. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{{\left(\left(1 + e^{0 - t}\right) \cdot \frac{1}{1 + e^{0 - s}}\right)}^{c\_p}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-240}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \mathsf{fma}\left(s, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, e^{0 - t}, 0.5\right)\right)}^{c\_p}\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= t -5e-240)
   (pow (/ 1.0 (fma s (fma s (fma s -0.16666666666666666 0.5) -1.0) 2.0)) c_p)
   (pow (fma 0.5 (exp (- 0.0 t)) 0.5) c_p)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -5e-240) {
		tmp = pow((1.0 / fma(s, fma(s, fma(s, -0.16666666666666666, 0.5), -1.0), 2.0)), c_p);
	} else {
		tmp = pow(fma(0.5, exp((0.0 - t)), 0.5), c_p);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (t <= -5e-240)
		tmp = Float64(1.0 / fma(s, fma(s, fma(s, -0.16666666666666666, 0.5), -1.0), 2.0)) ^ c_p;
	else
		tmp = fma(0.5, exp(Float64(0.0 - t)), 0.5) ^ c_p;
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[t, -5e-240], N[Power[N[(1.0 / N[(s * N[(s * N[(s * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[Power[N[(0.5 * N[Exp[N[(0.0 - t), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision], c$95$p], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.0000000000000004e-240

   1. Initial program 86.6%

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

   3. Taylor expanded in c_p around 0

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

   5. Simplified 93.9%

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

   6. Taylor expanded in c_n around 0

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

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

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

      2. neg-mul-1 N/A

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

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

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

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

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

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

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

      6. neg-mul-1 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 92.1

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

   8. Simplified 92.1%

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

   9. Taylor expanded in s around 0

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

       1. +-commutative N/A

          \[\leadsto {\left(\frac{1}{\color{blue}{s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right) + 2}}\right)}^{c\_p} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto {\left(\frac{1}{\color{blue}{\mathsf{fma}\left(s, s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1, 2\right)}}\right)}^{c\_p} \]

       3. sub-neg N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}\right)}^{c\_p} \]

       4. metadata-eval N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) + \color{blue}{-1}, 2\right)}\right)}^{c\_p} \]

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

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(s, \frac{1}{2} + \frac{-1}{6} \cdot s, -1\right)}, 2\right)}\right)}^{c\_p} \]

       6. +-commutative N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{\frac{-1}{6} \cdot s + \frac{1}{2}}, -1\right), 2\right)}\right)}^{c\_p} \]

       7. *-commutative N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{s \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 2\right)}\right)}^{c\_p} \]

       8. accelerator-lowering-fma.f64 96.1

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

   11. Simplified 96.1%

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

   if -5.0000000000000004e-240 < t

   1. Initial program 91.3%

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

   3. Taylor expanded in c_n around 0

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

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

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

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

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

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

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

      6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. --lowering--.f64 93.8

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

   5. Simplified 93.8%

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

   7. Applied egg-rr 96.4%

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

   8. Taylor expanded in s around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot \left(1 + e^{\mathsf{neg}\left(t\right)}\right)\right)}}^{c\_p} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {\left(\frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(t\right)} + 1\right)}\right)}^{c\_p} \]

      2. distribute-lft-in N/A

         \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \frac{1}{2} \cdot 1\right)}}^{c\_p} \]

      3. metadata-eval N/A

         \[\leadsto {\left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \color{blue}{\frac{1}{2}}\right)}^{c\_p} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, e^{\mathsf{neg}\left(t\right)}, \frac{1}{2}\right)\right)}}^{c\_p} \]

      5. neg-mul-1 N/A

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

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

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

      7. neg-mul-1 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{\mathsf{neg}\left(t\right)}}, \frac{1}{2}\right)\right)}^{c\_p} \]

      8. neg-sub0 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{0 - t}}, \frac{1}{2}\right)\right)}^{c\_p} \]

      9. --lowering--.f64 97.6

         \[\leadsto {\left(\mathsf{fma}\left(0.5, e^{\color{blue}{0 - t}}, 0.5\right)\right)}^{c\_p} \]

   10. Simplified 97.6%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(0.5, e^{0 - t}, 0.5\right)\right)}}^{c\_p} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-240}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \mathsf{fma}\left(s, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, c\_p \cdot t, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= t -5e-240)
   (pow (/ 1.0 (fma s (fma s (fma s -0.16666666666666666 0.5) -1.0) 2.0)) c_p)
   (fma -0.5 (* c_p t) 1.0)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -5e-240) {
		tmp = pow((1.0 / fma(s, fma(s, fma(s, -0.16666666666666666, 0.5), -1.0), 2.0)), c_p);
	} else {
		tmp = fma(-0.5, (c_p * t), 1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (t <= -5e-240)
		tmp = Float64(1.0 / fma(s, fma(s, fma(s, -0.16666666666666666, 0.5), -1.0), 2.0)) ^ c_p;
	else
		tmp = fma(-0.5, Float64(c_p * t), 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[t, -5e-240], N[Power[N[(1.0 / N[(s * N[(s * N[(s * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[(-0.5 * N[(c$95$p * t), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.0000000000000004e-240

   1. Initial program 86.6%

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

   3. Taylor expanded in c_p around 0

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

   5. Simplified 93.9%

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

   6. Taylor expanded in c_n around 0

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

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

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

      2. neg-mul-1 N/A

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

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

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

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

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

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

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

      6. neg-mul-1 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 92.1

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

   8. Simplified 92.1%

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

   9. Taylor expanded in s around 0

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

       1. +-commutative N/A

          \[\leadsto {\left(\frac{1}{\color{blue}{s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right) + 2}}\right)}^{c\_p} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto {\left(\frac{1}{\color{blue}{\mathsf{fma}\left(s, s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1, 2\right)}}\right)}^{c\_p} \]

       3. sub-neg N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}\right)}^{c\_p} \]

       4. metadata-eval N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) + \color{blue}{-1}, 2\right)}\right)}^{c\_p} \]

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

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(s, \frac{1}{2} + \frac{-1}{6} \cdot s, -1\right)}, 2\right)}\right)}^{c\_p} \]

       6. +-commutative N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{\frac{-1}{6} \cdot s + \frac{1}{2}}, -1\right), 2\right)}\right)}^{c\_p} \]

       7. *-commutative N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{s \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 2\right)}\right)}^{c\_p} \]

       8. accelerator-lowering-fma.f64 96.1

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

   11. Simplified 96.1%

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

   if -5.0000000000000004e-240 < t

   1. Initial program 91.3%

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

   3. Taylor expanded in c_n around 0

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

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

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

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

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

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

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

      6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. --lowering--.f64 93.8

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

   5. Simplified 93.8%

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

   7. Applied egg-rr 96.4%

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

   8. Taylor expanded in s around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot \left(1 + e^{\mathsf{neg}\left(t\right)}\right)\right)}}^{c\_p} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {\left(\frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(t\right)} + 1\right)}\right)}^{c\_p} \]

      2. distribute-lft-in N/A

         \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \frac{1}{2} \cdot 1\right)}}^{c\_p} \]

      3. metadata-eval N/A

         \[\leadsto {\left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \color{blue}{\frac{1}{2}}\right)}^{c\_p} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, e^{\mathsf{neg}\left(t\right)}, \frac{1}{2}\right)\right)}}^{c\_p} \]

      5. neg-mul-1 N/A

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

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

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

      7. neg-mul-1 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{\mathsf{neg}\left(t\right)}}, \frac{1}{2}\right)\right)}^{c\_p} \]

      8. neg-sub0 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{0 - t}}, \frac{1}{2}\right)\right)}^{c\_p} \]

      9. --lowering--.f64 97.6

         \[\leadsto {\left(\mathsf{fma}\left(0.5, e^{\color{blue}{0 - t}}, 0.5\right)\right)}^{c\_p} \]

   10. Simplified 97.6%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(0.5, e^{0 - t}, 0.5\right)\right)}}^{c\_p} \]

   11. Taylor expanded in t around 0

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

       1. associate-*r* N/A

          \[\leadsto 1 + \color{blue}{\left(\frac{-1}{2} \cdot c\_p\right) \cdot t} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot c\_p\right) \cdot t + 1} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(c\_p \cdot t\right)} + 1 \]

       4. accelerator-lowering-fma.f64 N/A

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

       5. *-lowering-*.f64 97.0

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

   13. Simplified 97.0%

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

Alternative 5: 94.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-241}:\\ \;\;\;\;{\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, 0.5, -1\right), 2\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, c\_p \cdot t, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= t -4e-241)
   (pow (/ 1.0 (fma s (fma s 0.5 -1.0) 2.0)) c_p)
   (fma -0.5 (* c_p t) 1.0)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (t <= -4e-241) {
		tmp = pow((1.0 / fma(s, fma(s, 0.5, -1.0), 2.0)), c_p);
	} else {
		tmp = fma(-0.5, (c_p * t), 1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (t <= -4e-241)
		tmp = Float64(1.0 / fma(s, fma(s, 0.5, -1.0), 2.0)) ^ c_p;
	else
		tmp = fma(-0.5, Float64(c_p * t), 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[t, -4e-241], N[Power[N[(1.0 / N[(s * N[(s * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[(-0.5 * N[(c$95$p * t), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.9999999999999999e-241

   1. Initial program 86.6%

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

   3. Taylor expanded in c_p around 0

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

   5. Simplified 93.9%

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

   6. Taylor expanded in c_n around 0

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

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

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

      2. neg-mul-1 N/A

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

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

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

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

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

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

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

      6. neg-mul-1 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 92.1

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

   8. Simplified 92.1%

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

   9. Taylor expanded in s around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. accelerator-lowering-fma.f64 95.1

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

   11. Simplified 95.1%

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

   if -3.9999999999999999e-241 < t

   1. Initial program 91.3%

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

   3. Taylor expanded in c_n around 0

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

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

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

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

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

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

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

      6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. --lowering--.f64 93.8

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

   5. Simplified 93.8%

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

   7. Applied egg-rr 96.4%

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

   8. Taylor expanded in s around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot \left(1 + e^{\mathsf{neg}\left(t\right)}\right)\right)}}^{c\_p} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {\left(\frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(t\right)} + 1\right)}\right)}^{c\_p} \]

      2. distribute-lft-in N/A

         \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \frac{1}{2} \cdot 1\right)}}^{c\_p} \]

      3. metadata-eval N/A

         \[\leadsto {\left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \color{blue}{\frac{1}{2}}\right)}^{c\_p} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, e^{\mathsf{neg}\left(t\right)}, \frac{1}{2}\right)\right)}}^{c\_p} \]

      5. neg-mul-1 N/A

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

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

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

      7. neg-mul-1 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{\mathsf{neg}\left(t\right)}}, \frac{1}{2}\right)\right)}^{c\_p} \]

      8. neg-sub0 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{0 - t}}, \frac{1}{2}\right)\right)}^{c\_p} \]

      9. --lowering--.f64 97.6

         \[\leadsto {\left(\mathsf{fma}\left(0.5, e^{\color{blue}{0 - t}}, 0.5\right)\right)}^{c\_p} \]

   10. Simplified 97.6%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(0.5, e^{0 - t}, 0.5\right)\right)}}^{c\_p} \]

   11. Taylor expanded in t around 0

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

       1. associate-*r* N/A

          \[\leadsto 1 + \color{blue}{\left(\frac{-1}{2} \cdot c\_p\right) \cdot t} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot c\_p\right) \cdot t + 1} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(c\_p \cdot t\right)} + 1 \]

       4. accelerator-lowering-fma.f64 N/A

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

       5. *-lowering-*.f64 97.0

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

   13. Simplified 97.0%

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

Alternative 6: 96.5% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq -155000000:\\ \;\;\;\;{\left(\frac{-6}{s \cdot \left(s \cdot s\right)}\right)}^{c\_p}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, c\_p \cdot t, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c_p c_n t s)
 :precision binary64
 (if (<= s -155000000.0)
   (pow (/ -6.0 (* s (* s s))) c_p)
   (fma -0.5 (* c_p t) 1.0)))

C

double code(double c_p, double c_n, double t, double s) {
	double tmp;
	if (s <= -155000000.0) {
		tmp = pow((-6.0 / (s * (s * s))), c_p);
	} else {
		tmp = fma(-0.5, (c_p * t), 1.0);
	}
	return tmp;
}

Julia

function code(c_p, c_n, t, s)
	tmp = 0.0
	if (s <= -155000000.0)
		tmp = Float64(-6.0 / Float64(s * Float64(s * s))) ^ c_p;
	else
		tmp = fma(-0.5, Float64(c_p * t), 1.0);
	end
	return tmp
end

Wolfram

code[c$95$p_, c$95$n_, t_, s_] := If[LessEqual[s, -155000000.0], N[Power[N[(-6.0 / N[(s * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], c$95$p], $MachinePrecision], N[(-0.5 * N[(c$95$p * t), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < -1.55e8

   1. Initial program 40.0%

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

   3. Taylor expanded in c_p around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in c_n around 0

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

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

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

      2. neg-mul-1 N/A

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

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

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

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

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

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

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

      6. neg-mul-1 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in s around 0

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

       1. +-commutative N/A

          \[\leadsto {\left(\frac{1}{\color{blue}{s \cdot \left(s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1\right) + 2}}\right)}^{c\_p} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto {\left(\frac{1}{\color{blue}{\mathsf{fma}\left(s, s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) - 1, 2\right)}}\right)}^{c\_p} \]

       3. sub-neg N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) + \left(\mathsf{neg}\left(1\right)\right)}, 2\right)}\right)}^{c\_p} \]

       4. metadata-eval N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, s \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot s\right) + \color{blue}{-1}, 2\right)}\right)}^{c\_p} \]

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

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \color{blue}{\mathsf{fma}\left(s, \frac{1}{2} + \frac{-1}{6} \cdot s, -1\right)}, 2\right)}\right)}^{c\_p} \]

       6. +-commutative N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{\frac{-1}{6} \cdot s + \frac{1}{2}}, -1\right), 2\right)}\right)}^{c\_p} \]

       7. *-commutative N/A

          \[\leadsto {\left(\frac{1}{\mathsf{fma}\left(s, \mathsf{fma}\left(s, \color{blue}{s \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 2\right)}\right)}^{c\_p} \]

       8. accelerator-lowering-fma.f64 100.0

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

   11. Simplified 100.0%

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

   12. Taylor expanded in s around inf

       \[\leadsto {\color{blue}{\left(\frac{-6}{{s}^{3}}\right)}}^{c\_p} \]
   13. Step-by-step derivation

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

          \[\leadsto {\color{blue}{\left(\frac{-6}{{s}^{3}}\right)}}^{c\_p} \]

       2. cube-mult N/A

          \[\leadsto {\left(\frac{-6}{\color{blue}{s \cdot \left(s \cdot s\right)}}\right)}^{c\_p} \]

       3. unpow2 N/A

          \[\leadsto {\left(\frac{-6}{s \cdot \color{blue}{{s}^{2}}}\right)}^{c\_p} \]

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

          \[\leadsto {\left(\frac{-6}{\color{blue}{s \cdot {s}^{2}}}\right)}^{c\_p} \]

       5. unpow2 N/A

          \[\leadsto {\left(\frac{-6}{s \cdot \color{blue}{\left(s \cdot s\right)}}\right)}^{c\_p} \]

       6. *-lowering-*.f64 100.0

          \[\leadsto {\left(\frac{-6}{s \cdot \color{blue}{\left(s \cdot s\right)}}\right)}^{c\_p} \]

   14. Simplified 100.0%

       \[\leadsto {\color{blue}{\left(\frac{-6}{s \cdot \left(s \cdot s\right)}\right)}}^{c\_p} \]

   if -1.55e8 < s

   1. Initial program 90.5%

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

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

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

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

      6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

      12. neg-sub0 N/A

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

      13. --lowering--.f64 92.6

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

   5. Simplified 92.6%

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

   7. Applied egg-rr 94.2%

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

   8. Taylor expanded in s around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot \left(1 + e^{\mathsf{neg}\left(t\right)}\right)\right)}}^{c\_p} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {\left(\frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(t\right)} + 1\right)}\right)}^{c\_p} \]

      2. distribute-lft-in N/A

         \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \frac{1}{2} \cdot 1\right)}}^{c\_p} \]

      3. metadata-eval N/A

         \[\leadsto {\left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \color{blue}{\frac{1}{2}}\right)}^{c\_p} \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, e^{\mathsf{neg}\left(t\right)}, \frac{1}{2}\right)\right)}}^{c\_p} \]

      5. neg-mul-1 N/A

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

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

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

      7. neg-mul-1 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{\mathsf{neg}\left(t\right)}}, \frac{1}{2}\right)\right)}^{c\_p} \]

      8. neg-sub0 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{0 - t}}, \frac{1}{2}\right)\right)}^{c\_p} \]

      9. --lowering--.f64 95.7

         \[\leadsto {\left(\mathsf{fma}\left(0.5, e^{\color{blue}{0 - t}}, 0.5\right)\right)}^{c\_p} \]

   10. Simplified 95.7%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(0.5, e^{0 - t}, 0.5\right)\right)}}^{c\_p} \]

   11. Taylor expanded in t around 0

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

       1. associate-*r* N/A

          \[\leadsto 1 + \color{blue}{\left(\frac{-1}{2} \cdot c\_p\right) \cdot t} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot c\_p\right) \cdot t + 1} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(c\_p \cdot t\right)} + 1 \]

       4. accelerator-lowering-fma.f64 N/A

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

       5. *-lowering-*.f64 96.1

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

   13. Simplified 96.1%

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

Alternative 7: 94.6% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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_n around 0

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

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

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

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

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

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

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

   12. neg-sub0 N/A

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

   13. --lowering--.f64 91.6

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

5. Simplified 91.6%

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

7. Applied egg-rr 94.3%

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

8. Taylor expanded in s around 0

   \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot \left(1 + e^{\mathsf{neg}\left(t\right)}\right)\right)}}^{c\_p} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto {\left(\frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(t\right)} + 1\right)}\right)}^{c\_p} \]

   2. distribute-lft-in N/A

      \[\leadsto {\color{blue}{\left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \frac{1}{2} \cdot 1\right)}}^{c\_p} \]

   3. metadata-eval N/A

      \[\leadsto {\left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(t\right)} + \color{blue}{\frac{1}{2}}\right)}^{c\_p} \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, e^{\mathsf{neg}\left(t\right)}, \frac{1}{2}\right)\right)}}^{c\_p} \]

   5. neg-mul-1 N/A

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

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

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

   7. neg-mul-1 N/A

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{\mathsf{neg}\left(t\right)}}, \frac{1}{2}\right)\right)}^{c\_p} \]

   8. neg-sub0 N/A

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, e^{\color{blue}{0 - t}}, \frac{1}{2}\right)\right)}^{c\_p} \]

   9. --lowering--.f64 93.9

      \[\leadsto {\left(\mathsf{fma}\left(0.5, e^{\color{blue}{0 - t}}, 0.5\right)\right)}^{c\_p} \]

10. Simplified 93.9%

    \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(0.5, e^{0 - t}, 0.5\right)\right)}}^{c\_p} \]

11. Taylor expanded in t around 0

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

    1. associate-*r* N/A

       \[\leadsto 1 + \color{blue}{\left(\frac{-1}{2} \cdot c\_p\right) \cdot t} \]

    2. +-commutative N/A

       \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot c\_p\right) \cdot t + 1} \]

    3. associate-*r* N/A

       \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(c\_p \cdot t\right)} + 1 \]

    4. accelerator-lowering-fma.f64 N/A

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

    5. *-lowering-*.f64 94.3

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

13. Simplified 94.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, c\_p \cdot t, 1\right)} \]
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 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_n around 0

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

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

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

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

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

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

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

   12. neg-sub0 N/A

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

   13. --lowering--.f64 91.6

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

5. Simplified 91.6%

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

6. Taylor expanded in c_p around 0

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

8. Simplified 94.2%

   \[\leadsto \color{blue}{1} \]
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 2024197 
(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))))