Average Error: 0.3 → 0.3
Time: 11.8s
Precision: binary32
\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]
\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
↓
\[\begin{array}{l} t_0 := 1 + e^{\frac{\pi}{s}}\\ \log \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{\frac{-\pi}{s}}} + \frac{-1}{t_0}, \frac{1}{t_0}\right)} + -1\right) \cdot \left(-s\right) \end{array} \]
(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (+
      (*
       u
       (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s))))))
      (/ 1.0 (+ 1.0 (exp (/ PI s))))))
    1.0))))
↓
(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (exp (/ PI s)))))
   (*
    (log
     (+
      (/
       1.0
       (fma u (+ (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ -1.0 t_0)) (/ 1.0 t_0)))
      -1.0))
    (- s))))
float code(float u, float s) {
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - (1.0f / (1.0f + expf((((float) M_PI) / s)))))) + (1.0f / (1.0f + expf((((float) M_PI) / s)))))) - 1.0f));
}
↓
float code(float u, float s) {
	float t_0 = 1.0f + expf((((float) M_PI) / s));
	return logf(((1.0f / fmaf(u, ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) + (-1.0f / t_0)), (1.0f / t_0))) + -1.0f)) * -s;
}

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))) + Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))) - Float32(1.0))))
end
↓
function code(u, s)
	t_0 = Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))
	return Float32(log(Float32(Float32(Float32(1.0) / fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) + Float32(Float32(-1.0) / t_0)), Float32(Float32(1.0) / t_0))) + Float32(-1.0))) * Float32(-s))
end

\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)
↓
\begin{array}{l}
t_0 := 1 + e^{\frac{\pi}{s}}\\
\log \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{\frac{-\pi}{s}}} + \frac{-1}{t_0}, \frac{1}{t_0}\right)} + -1\right) \cdot \left(-s\right)
\end{array}

Error

Bits error versus u

Bits error versus s

Derivation

  1. Initial program 0.3

    \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

  2. Applied egg-rr0.8

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + -1\right)}\right)}^{3}} \]

  3. Applied egg-rr0.3

    \[\leadsto \color{blue}{\log \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + -1\right) \cdot \left(-s\right)} \]

  4. Final simplification0.3

    \[\leadsto \log \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{\frac{-\pi}{s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + -1\right) \cdot \left(-s\right) \]

Reproduce

herbie shell --seed 2022153 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) 1.0))))