Logistic distribution

Average Error: 0.2 → 0.2

Time: 11.8s

Precision: binary32

Cost: 6848

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

Math

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

↓

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

FPCore

(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 (+ 2.0 (+ (exp (/ x s)) (exp (/ (- x) s))))) s))

C

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 / (2.0f + (expf((x / s)) + expf((-x / s))))) / s;
}

Fortran

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 / (2.0e0 + (exp((x / s)) + exp((-x / s))))) / s
end function

Julia

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

MATLAB

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) / (single(2.0) + (exp((x / s)) + exp((-x / s))))) / s;
end

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. Simplified 0.2

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}} \]
   Proof
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) (neg.f32 s))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) (Rewrite<= mul-1-neg_binary32 (*.f32 -1 s)))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (Rewrite=> associate-/r*_binary32 (/.f32 (/.f32 (fabs.f32 x) -1) s))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (/.f32 (/.f32 (fabs.f32 x) (Rewrite<= metadata-eval (/.f32 1 -1))) s)) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (/.f32 (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 (fabs.f32 x) -1) 1)) s)) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (/.f32 (/.f32 (Rewrite<= *-commutative_binary32 (*.f32 -1 (fabs.f32 x))) 1) s)) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (/.f32 (/.f32 (Rewrite<= neg-mul-1_binary32 (neg.f32 (fabs.f32 x))) 1) s)) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (/.f32 (Rewrite=> /-rgt-identity_binary32 (neg.f32 (fabs.f32 x))) s)) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (Rewrite<= +-lft-identity_binary32 (+.f32 0 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (+.f32 (Rewrite<= +-inverses_binary32 (-.f32 (/.f32 (fabs.f32 x) s) (/.f32 (fabs.f32 x) s))) (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 64 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (+.f32 (Rewrite<= unsub-neg_binary32 (+.f32 (/.f32 (fabs.f32 x) s) (neg.f32 (/.f32 (fabs.f32 x) s)))) (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (+.f32 (+.f32 (/.f32 (fabs.f32 x) s) (Rewrite<= distribute-frac-neg_binary32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (exp.f32 (Rewrite<= associate-+r+_binary32 (+.f32 (/.f32 (fabs.f32 x) s) (+.f32 (/.f32 (neg.f32 (fabs.f32 x)) s) (/.f32 (neg.f32 (fabs.f32 x)) s))))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (Rewrite<= prod-exp_binary32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (exp.f32 (+.f32 (/.f32 (neg.f32 (fabs.f32 x)) s) (/.f32 (neg.f32 (fabs.f32 x)) s))))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 123 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (Rewrite<= prod-exp_binary32 (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 2))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (Rewrite<= metadata-eval (+.f32 1 1))))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (Rewrite<= associate-+l+_binary32 (+.f32 (+.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 1) 1)))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (+.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 1)) 1) 1))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 1) (Rewrite<= rgt-mult-inverse_binary32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (/.f32 1 (exp.f32 (/.f32 (fabs.f32 x) s)))))) 1))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 1) (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (Rewrite<= exp-neg_binary32 (exp.f32 (neg.f32 (/.f32 (fabs.f32 x) s)))))) 1))): 1 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) 1) (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (exp.f32 (Rewrite<= distribute-frac-neg_binary32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) 1))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (Rewrite<= distribute-lft-in_binary32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) 1))): 0 points increase in error, 2 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (Rewrite<= *-commutative_binary32 (*.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (exp.f32 (/.f32 (fabs.f32 x) s)))) 1))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (Rewrite=> *-commutative_binary32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))) 1))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (Rewrite<= rgt-mult-inverse_binary32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (/.f32 1 (exp.f32 (/.f32 (fabs.f32 x) s)))))))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (Rewrite<= exp-neg_binary32 (exp.f32 (neg.f32 (/.f32 (fabs.f32 x) s)))))))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (exp.f32 (Rewrite<= distribute-frac-neg_binary32 (/.f32 (neg.f32 (fabs.f32 x)) s))))))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (+.f32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (Rewrite<= distribute-lft-in_binary32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (+.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))))): 3 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (Rewrite<= distribute-lft-in_binary32 (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (+.f32 (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))))): 4 points increase in error, 188 points decrease in error
   (/.f32 (/.f32 1 s) (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (Rewrite=> +-commutative_binary32 (+.f32 (+.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (Rewrite<= associate-+r+_binary32 (+.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))))): 4 points increase in error, 3 points decrease in error
   (/.f32 (/.f32 1 s) (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (+.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) 1)) (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (+.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (Rewrite<= distribute-lft-in_binary32 (*.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))))): 2 points increase in error, 3 points decrease in error
   (/.f32 (/.f32 1 s) (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (Rewrite=> distribute-rgt1-in_binary32 (*.f32 (+.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) 1) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))))): 1 points increase in error, 2 points decrease in error
   (/.f32 (/.f32 1 s) (*.f32 (exp.f32 (/.f32 (fabs.f32 x) s)) (*.f32 (Rewrite<= +-commutative_binary32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (Rewrite<= *-commutative_binary32 (*.f32 (*.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (exp.f32 (/.f32 (fabs.f32 x) s))))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (*.f32 (Rewrite<= /-rgt-identity_binary32 (/.f32 (*.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) 1)) (exp.f32 (/.f32 (fabs.f32 x) s)))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 1 s) (Rewrite<= associate-/r/_binary32 (/.f32 (*.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (/.f32 1 (exp.f32 (/.f32 (fabs.f32 x) s)))))): 2 points increase in error, 1 points decrease in error
   (/.f32 (/.f32 1 s) (/.f32 (*.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (Rewrite<= exp-neg_binary32 (exp.f32 (neg.f32 (/.f32 (fabs.f32 x) s)))))): 4 points increase in error, 2 points decrease in error
   (/.f32 (/.f32 1 s) (/.f32 (*.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (exp.f32 (Rewrite<= distribute-frac-neg_binary32 (/.f32 (neg.f32 (fabs.f32 x)) s))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 (/.f32 1 s) (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (*.f32 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))): 3 points increase in error, 5 points decrease in error
   (Rewrite=> times-frac_binary32 (*.f32 (/.f32 (/.f32 1 s) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))): 2 points increase in error, 6 points decrease in error
   (*.f32 (Rewrite<= associate-/r*_binary32 (/.f32 1 (*.f32 s (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))) (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))): 3 points increase in error, 2 points decrease in error
   (Rewrite<= times-frac_binary32 (/.f32 (*.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))) (*.f32 (*.f32 s (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))): 6 points increase in error, 4 points decrease in error
   (Rewrite<= associate-*r/_binary32 (*.f32 1 (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> *-lft-identity_binary32 (/.f32 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)) (*.f32 (*.f32 s (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (fabs.f32 x)) s)))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(\cosh \left(\frac{x}{s}\right), 2, 2\right)}}{s}} \]

5. Taylor expanded in x around inf 0.2

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

6. Simplified 0.2

   \[\leadsto \frac{\frac{1}{\color{blue}{2 + \left(e^{\frac{x}{s}} + e^{\frac{-x}{s}}\right)}}}{s} \]
   Proof
   (+.f32 2 (+.f32 (exp.f32 (/.f32 x s)) (exp.f32 (/.f32 (neg.f32 x) s)))): 0 points increase in error, 0 points decrease in error
   (+.f32 2 (+.f32 (exp.f32 (/.f32 x s)) (exp.f32 (Rewrite<= distribute-neg-frac_binary32 (neg.f32 (/.f32 x s)))))): 0 points increase in error, 0 points decrease in error
   (+.f32 2 (+.f32 (exp.f32 (/.f32 x s)) (Rewrite<= rec-exp_binary32 (/.f32 1 (exp.f32 (/.f32 x s)))))): 0 points increase in error, 2 points decrease in error
   (Rewrite<= associate-+l+_binary32 (+.f32 (+.f32 2 (exp.f32 (/.f32 x s))) (/.f32 1 (exp.f32 (/.f32 x s))))): 2 points increase in error, 1 points decrease in error
   (+.f32 (Rewrite<= +-commutative_binary32 (+.f32 (exp.f32 (/.f32 x s)) 2)) (/.f32 1 (exp.f32 (/.f32 x s)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary32 (+.f32 (exp.f32 (/.f32 x s)) (+.f32 2 (/.f32 1 (exp.f32 (/.f32 x s)))))): 0 points increase in error, 1 points decrease in error

7. Final simplification 0.2

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

Alternatives

Alternative 1
Error	3.3
Cost	3620

\[\begin{array}{l} \mathbf{if}\;x \leq 2.1999999787645047 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{1}{s}}{3 + \frac{1}{e^{\frac{x}{s}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{\frac{x}{\frac{s \cdot s}{x}} + 4}\\ \end{array} \]

Alternative 2
Error	3.3
Cost	3588

\[\begin{array}{l} \mathbf{if}\;x \leq 2.1999999787645047 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{-x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{\frac{x}{\frac{s \cdot s}{x}} + 4}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	3552

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

Alternative 4
Error	4.5
Cost	1000

\[\begin{array}{l} t_0 := \frac{0.25}{s \cdot s}\\ \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{elif}\;x \leq -1.4999999988791435 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{1}{s}}{3 + \left(\left(1 - \frac{x}{s}\right) + x \cdot \left(x \cdot \left(t_0 + t_0\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.1999999787645047 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{1}{4 + \frac{x}{s} \cdot \frac{x}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{\frac{x}{\frac{s \cdot s}{x}} + 4}\\ \end{array} \]

Alternative 5
Error	4.9
Cost	548

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{1}{\frac{x}{\frac{s \cdot s}{x}} + 4}\\ \end{array} \]

Alternative 6
Error	6.5
Cost	484

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4 + \frac{x}{s} \cdot \frac{x}{s}}}{s}\\ \end{array} \]

Alternative 7
Error	6.3
Cost	424

\[\begin{array}{l} t_0 := \left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{if}\;x \leq -5.999999941330714 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	12.1
Cost	360

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 9
Error	12.1
Cost	360

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;s \cdot \frac{-1}{x \cdot \left(-x\right)}\\ \mathbf{elif}\;x \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 10
Error	12.1
Cost	296

\[\begin{array}{l} t_0 := \frac{s}{x \cdot x}\\ \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	23.5
Cost	96

\[\frac{0.25}{s} \]

Reproduce

herbie shell --seed 2022300 
(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))))))