Logistic distribution

Average Error: 0.1 → 0.1

Time: 12.4s

Precision: binary32

Cost: 9952

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

Math

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

↓

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

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 (* s (+ 2.0 (* 2.0 (exp (log (cosh (/ x 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 / (s * (2.0f + (2.0f * expf(logf(coshf((x / 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 / (s * (2.0e0 + (2.0e0 * exp(log(cosh((x / 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(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * exp(log(cosh(Float32(x / 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) / (s * (single(2.0) + (single(2.0) * exp(log(cosh((x / s)))))));
end

Derivation

1. Initial program 0.1

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

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))): 70 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))): 115 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, 1 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, 1 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))): 3 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))): 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 (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))))))): 6 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))))))): 2 points increase in error, 185 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)))))))): 2 points increase in error, 5 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)))))))): 4 points increase in error, 2 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, 0 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, 3 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)))))): 7 points increase in error, 5 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)))))): 5 points increase in error, 9 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)))))): 5 points increase in error, 4 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))))): 8 points increase in error, 7 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)))))): 8 points increase in error, 9 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 0.8

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

4. Simplified 0.1

   \[\leadsto \color{blue}{\frac{1}{s \cdot \left(2 + 2 \cdot \cosh \left(\frac{x}{s}\right)\right)}} \]
   Proof
   (/.f32 1 (*.f32 s (+.f32 2 (*.f32 2 (cosh.f32 (/.f32 x s)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= expm1-log1p_binary32 (expm1.f32 (log1p.f32 (/.f32 1 (*.f32 s (+.f32 2 (*.f32 2 (cosh.f32 (/.f32 x s))))))))): 57 points increase in error, 1 points decrease in error
   (Rewrite<= expm1-def_binary32 (-.f32 (exp.f32 (log1p.f32 (/.f32 1 (*.f32 s (+.f32 2 (*.f32 2 (cosh.f32 (/.f32 x s)))))))) 1)): 7 points increase in error, 4 points decrease in error

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	3552

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

Alternative 2
Error	4.0
Cost	3428

\[\begin{array}{l} \mathbf{if}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s}\\ \mathbf{elif}\;x \leq 1.2000000104706032 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s \cdot \frac{s}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \end{array} \]

Alternative 3
Error	4.9
Cost	620

\[\begin{array}{l} t_0 := \left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\left(1 + s \cdot 4\right) + -1\\ \mathbf{elif}\;x \leq 1.2000000104706032 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	4.3
Cost	620

\[\begin{array}{l} t_0 := \left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\left(1 + s \cdot 4\right) + -1\\ \mathbf{elif}\;x \leq 1.2000000104706032 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s \cdot \frac{s}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	4.7
Cost	560

\[\begin{array}{l} t_0 := \left(1 + s \cdot 4\right) + -1\\ t_1 := \left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.3350000254891322 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 0.00019999999494757503:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	6.8
Cost	428

\[\begin{array}{l} t_0 := \left(1 + s \cdot 4\right) + -1\\ \mathbf{if}\;x \leq -1.3350000254891322 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 999999986991104:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 7
Error	6.8
Cost	428

\[\begin{array}{l} t_0 := \left(1 + s \cdot 4\right) + -1\\ \mathbf{if}\;x \leq -1.3350000254891322 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 999999986991104:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 8
Error	12.0
Cost	296

\[\begin{array}{l} t_0 := \frac{s}{x \cdot x}\\ \mathbf{if}\;x \leq -0.0006000000284984708:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	12.0
Cost	296

\[\begin{array}{l} t_0 := \frac{\frac{s}{x}}{x}\\ \mathbf{if}\;x \leq -0.0006000000284984708:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	28.5
Cost	96

\[s \cdot 4 \]

Alternative 11
Error	23.2
Cost	96

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

Alternative 12
Error	29.3
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022325 
(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))))))