?

Average Accuracy: 99.6% → 99.4%
Time: 15.2s
Precision: binary32
Cost: 13248

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)} \]
(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) (+ (exp (/ (fabs x) (- s))) (+ (exp (/ (fabs x) s)) 2.0))))
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) / (expf((fabsf(x) / -s)) + (expf((fabsf(x) / s)) + 2.0f));
}

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

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(Float32(1.0) / s) / Float32(exp(Float32(abs(x) / Float32(-s))) + Float32(exp(Float32(abs(x) / s)) + Float32(2.0))))
end

function tmp = code(x, s)
	tmp = exp((-abs(x) / s)) / ((s * (single(1.0) + exp((-abs(x) / s)))) * (single(1.0) + exp((-abs(x) / s))));
end
↓
function tmp = code(x, s)
	tmp = (single(1.0) / s) / (exp((abs(x) / -s)) + (exp((abs(x) / s)) + single(2.0)));
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{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.6%

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

  2. Simplified99.4%

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

    [Start]99.6

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

    associate-*l* [=>]99.6

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

    associate-/r* [=>]99.6

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

    distribute-frac-neg [=>]99.6

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

    exp-neg [=>]99.5

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

    associate-/l/ [=>]99.5

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

    associate-/r* [=>]99.4

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

    associate-/r* [<=]99.4

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

  3. Final simplification99.4%

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

Alternatives

Alternative 1
Accuracy99.6%
Cost6848
\[\frac{1}{s \cdot \left(e^{\frac{-x}{s}} + \left(2 + e^{\frac{x}{s}}\right)\right)} \]
Alternative 2
Accuracy99.4%
Cost6848
\[\frac{\frac{1}{s}}{e^{\frac{x}{s}} + \left(2 + e^{\frac{-x}{s}}\right)} \]
Alternative 3
Accuracy96.1%
Cost6752
\[\frac{\frac{1}{s}}{1 + \left(e^{\frac{\left|x\right|}{s}} + 2\right)} \]
Alternative 4
Accuracy96.7%
Cost6752
\[\frac{1}{s \cdot \left(e^{\frac{\left|x\right|}{s}} + 3\right) - x} \]
Alternative 5
Accuracy94.7%
Cost6656
\[\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4} \]
Alternative 6
Accuracy78.3%
Cost3624
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;1 + \left(-1 + s \cdot 4\right)\\ \mathbf{elif}\;x \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(4, s, \frac{x \cdot x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{s}}{-3 + \left(\left(-1 - \frac{x}{s}\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{s \cdot s}\right)}\\ \end{array} \]
Alternative 7
Accuracy78.7%
Cost3624
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;1 + \left(-1 + s \cdot 4\right)\\ \mathbf{elif}\;x \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(s, 4, \frac{x}{\frac{s}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{s}}{-3 + \left(\left(-1 - \frac{x}{s}\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{s \cdot s}\right)}\\ \end{array} \]
Alternative 8
Accuracy95.6%
Cost3620
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + \frac{1}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{s}}{-3 - t_0}\\ \end{array} \]
Alternative 9
Accuracy95.5%
Cost3588
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{-x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{s}}{-3 - e^{\frac{x}{s}}}\\ \end{array} \]
Alternative 10
Accuracy86.0%
Cost3556
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;1 + \left(-1 + s \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{s}}{-3 - e^{\frac{x}{s}}}\\ \end{array} \]
Alternative 11
Accuracy78.6%
Cost808
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;1 + \left(-1 + s \cdot 4\right)\\ \mathbf{elif}\;x \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{s}}{-3 + \left(\left(-1 - \frac{x}{s}\right) + \frac{\left(x \cdot x\right) \cdot -0.5}{s \cdot s}\right)}\\ \end{array} \]
Alternative 12
Accuracy75.9%
Cost361
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20} \lor \neg \left(x \leq 5.000000136226006 \cdot 10^{-28}\right):\\ \;\;\;\;1 + \left(-1 + s \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 13
Accuracy76.0%
Cost360
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;1 + \left(-1 + s \cdot 4\right)\\ \mathbf{elif}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + s \cdot 4\right)\\ \end{array} \]
Alternative 14
Accuracy11.0%
Cost96
\[s \cdot 4 \]
Alternative 15
Accuracy27.1%
Cost96
\[\frac{0.25}{s} \]
Alternative 16
Accuracy8.3%
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023122 
(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))))))