Average Error: 0.2 → 0.1
Time: 5.3s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{\frac{1}{e^{\frac{-\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)}}{s} \]
(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 (+ (exp (/ (- (fabs x)) s)) (+ 2.0 (exp (/ (fabs x) s))))) 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 / (expf((-fabsf(x) / s)) + (2.0f + expf((fabsf(x) / s))))) / s;
}

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 / (exp((-abs(x) / s)) + (2.0e0 + exp((abs(x) / s))))) / s
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) / Float32(exp(Float32(Float32(-abs(x)) / s)) + Float32(Float32(2.0) + exp(Float32(abs(x) / s))))) / s)
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) / (exp((-abs(x) / s)) + (single(2.0) + exp((abs(x) / s))))) / 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{\frac{1}{e^{\frac{-\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)}}{s}

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. Simplified0.1

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

  3. Taylor expanded in x around 0 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022211 
(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))))))