Average Error: 0.2 → 0.2
Time: 10.3s
Precision: binary32
Cost: 9952
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{1}{s \cdot \left(2 + 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cosh \left(\frac{x}{s}\right)\right)\right)\right)} \]
(FPCore (x s)
 :precision binary32
 (/
  (exp (/ (- (fabs x)) s))
  (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))
↓
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (+ 2.0 (* 2.0 (expm1 (log1p (cosh (/ x s)))))))))
float code(float x, float s) {
	return expf((-fabsf(x) / s)) / ((s * (1.0f + expf((-fabsf(x) / s)))) * (1.0f + expf((-fabsf(x) / s))));
}
↓
float code(float x, float s) {
	return 1.0f / (s * (2.0f + (2.0f * expm1f(log1pf(coshf((x / s)))))));
}

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(s * Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s)))) * Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s)))))
end
↓
function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * expm1(log1p(cosh(Float32(x / s))))))))
end

\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}
↓
\frac{1}{s \cdot \left(2 + 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cosh \left(\frac{x}{s}\right)\right)\right)\right)}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Applied egg-rr0.2

    \[\leadsto \color{blue}{\frac{1}{s \cdot \left(2 + 2 \cdot \cosh \left(\frac{x}{s}\right)\right)} \cdot 1} \]

  4. Applied egg-rr0.2

    \[\leadsto \frac{1}{s \cdot \left(2 + 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh \left(\frac{x}{s}\right)\right)\right)}\right)} \cdot 1 \]

  5. Final simplification0.2

    \[\leadsto \frac{1}{s \cdot \left(2 + 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cosh \left(\frac{x}{s}\right)\right)\right)\right)} \]

Alternatives

Alternative 1
Error0.1
Cost6752
\[\frac{1}{\mathsf{fma}\left(\cosh \left(\frac{x}{s}\right), s \cdot 2, s \cdot 2\right)} \]
Alternative 2
Error3.4
Cost3620
\[\begin{array}{l} \mathbf{if}\;x \leq 2.800000062604757 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{e^{\frac{x}{s}}}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{s \cdot \left(2 + 2 \cdot \cosh \left(\frac{x}{s}\right)\right)}\\ \end{array} \]
Alternative 3
Error0.2
Cost3616
\[\frac{1}{2 + 2 \cdot \cosh \left(\frac{x}{s}\right)} \cdot \frac{1}{s} \]
Alternative 4
Error0.2
Cost3616
\[\frac{1}{\frac{s}{\frac{1}{2 + 2 \cdot \cosh \left(\frac{x}{s}\right)}}} \]
Alternative 5
Error4.8
Cost3492
\[\begin{array}{l} \mathbf{if}\;x \leq -6.000000019026461 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{e^{\frac{x}{s}}}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \end{array} \]
Alternative 6
Error6.0
Cost3364
\[\begin{array}{l} \mathbf{if}\;x \leq -9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;e^{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \end{array} \]
Alternative 7
Error8.9
Cost552
\[\begin{array}{l} t_0 := \frac{-1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \mathbf{if}\;x \leq -9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{0.25 - \frac{x}{s} \cdot \frac{x}{\frac{s}{0.0625}}}{s}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error11.4
Cost488
\[\begin{array}{l} t_0 := \frac{-1}{s \cdot 4 + \frac{x}{\frac{s}{x}}}\\ \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error7.5
Cost480
\[\frac{1}{\frac{s}{\frac{1}{4 + \frac{x}{s} \cdot \frac{x}{s}}}} \]
Alternative 10
Error7.5
Cost416
\[\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)} \]
Alternative 11
Error11.8
Cost328
\[\begin{array}{l} t_0 := \frac{-s}{x \cdot x}\\ \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 12
Error23.2
Cost96
\[\frac{0.25}{s} \]

Error

Reproduce

herbie shell --seed 2022228 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))