Logistic function

Average Accuracy: 99.8% → 99.8%

Time: 12.3s

Precision: binary32

Cost: 16416.00

\[0 \leq s \land s \leq 1.0651631\]

Math

\[\frac{1}{1 + e^{\frac{-x}{s}}} \]

↓

\[\frac{1}{1 + \frac{{\left({\left(e^{-0.25}\right)}^{\left(\frac{x}{s}\right)}\right)}^{2}}{\sqrt{e^{\frac{x}{s}}}}} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))

↓

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (/ (pow (pow (exp -0.25) (/ x s)) 2.0) (sqrt (exp (/ x s)))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}

↓

float code(float x, float s) {
	return 1.0f / (1.0f + (powf(powf(expf(-0.25f), (x / s)), 2.0f) / sqrtf(expf((x / s)))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + exp((-x / s)))
end function

↓

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (((exp((-0.25e0)) ** (x / s)) ** 2.0e0) / sqrt(exp((x / s)))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
end

↓

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(((exp(Float32(-0.25)) ^ Float32(x / s)) ^ Float32(2.0)) / sqrt(exp(Float32(x / s))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + exp((-x / s)));
end

↓

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (((exp(single(-0.25)) ^ (x / s)) ^ single(2.0)) / sqrt(exp((x / s)))));
end

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]

2. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{\frac{1}{\sqrt{e^{\frac{x}{s}}}}}{\sqrt{e^{\frac{x}{s}}}}}} \]

3. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \frac{\color{blue}{{\left({\left(\sqrt{e^{\frac{x}{s}}}\right)}^{-0.5}\right)}^{2}}}{\sqrt{e^{\frac{x}{s}}}}} \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{1 + \frac{{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s} \cdot -0.25}\right)} - 1\right)}}^{2}}{\sqrt{e^{\frac{x}{s}}}}} \]

5. Simplified 99.8%

   \[\leadsto \frac{1}{1 + \frac{{\color{blue}{\left({\left(e^{-0.25}\right)}^{\left(\frac{x}{s}\right)}\right)}}^{2}}{\sqrt{e^{\frac{x}{s}}}}} \]
   Proof

   [Start] 99.7	\[ \frac{1}{1 + \frac{{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s} \cdot -0.25}\right)} - 1\right)}^{2}}{\sqrt{e^{\frac{x}{s}}}}} \]
   expm1-def [=>] 99.7	\[ \frac{1}{1 + \frac{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s} \cdot -0.25}\right)\right)\right)}}^{2}}{\sqrt{e^{\frac{x}{s}}}}} \]
   expm1-log1p [=>] 99.8	\[ \frac{1}{1 + \frac{{\color{blue}{\left(e^{\frac{x}{s} \cdot -0.25}\right)}}^{2}}{\sqrt{e^{\frac{x}{s}}}}} \]
   *-commutative [<=] 99.8	\[ \frac{1}{1 + \frac{{\left(e^{\color{blue}{-0.25 \cdot \frac{x}{s}}}\right)}^{2}}{\sqrt{e^{\frac{x}{s}}}}} \]
   exp-prod [=>] 99.8	\[ \frac{1}{1 + \frac{{\color{blue}{\left({\left(e^{-0.25}\right)}^{\left(\frac{x}{s}\right)}\right)}}^{2}}{\sqrt{e^{\frac{x}{s}}}}} \]

6. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + \frac{{\left({\left(e^{-0.25}\right)}^{\left(\frac{x}{s}\right)}\right)}^{2}}{\sqrt{e^{\frac{x}{s}}}}} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13184.00

\[\frac{1}{1 + \frac{{\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}}{\sqrt{e^{\frac{x}{s}}}}} \]

Alternative 2
Accuracy	99.8%
Cost	3456.00

\[\frac{1}{1 + e^{\frac{-x}{s}}} \]

Alternative 3
Accuracy	91.9%
Cost	744.00

\[\begin{array}{l} \mathbf{if}\;x \leq -1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\left(1 + \frac{s}{x}\right) + -1\\ \mathbf{elif}\;x \leq -5.00000011871114 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{1 + \left(\left(1 - x \cdot \left(x \cdot \frac{-0.5}{s \cdot s}\right)\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 4
Accuracy	75.3%
Cost	552.00

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]

Alternative 5
Accuracy	75.3%
Cost	552.00

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;0.5 + \frac{0.25}{\frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]

Alternative 6
Accuracy	92.7%
Cost	552.00

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;0.5 + \frac{0.25}{\frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \end{array} \]

Alternative 7
Accuracy	92.5%
Cost	552.00

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;0.5 + \frac{0.25}{\frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{s}{x}\right) + -1\\ \end{array} \]

Alternative 8
Accuracy	92.5%
Cost	552.00

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ t_1 := 1 + \frac{s}{x}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;\frac{1}{t_1}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;0.5 + \frac{0.25}{\frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1 + -1\\ \end{array} \]

Alternative 9
Accuracy	73.2%
Cost	520.00

\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]

Alternative 10
Accuracy	92.0%
Cost	516.00

\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{s}{x}\right) + -1\\ \end{array} \]

Alternative 11
Accuracy	67.9%
Cost	296.00

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 12
Accuracy	46.4%
Cost	164.00

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 13
Accuracy	35.5%
Cost	32.00

\[0.5 \]

Reproduce

herbie shell --seed 2023096 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))