Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 12.8s

Alternatives: 7

Speedup: N/A×

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / 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)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) t_0)) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	return -s * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - t_0)) + t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / 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)))) - t_0)) + t_0)) - Float32(1.0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) / (single(1.0) + exp((single(pi) / s)));
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - t_0)) + t_0)) - single(1.0)));
end

Alternative 1: 99.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))))
     -1.0)))))

C

float code(float u, float s) {
	return s * -logf(((1.0f / ((u / (1.0f + expf((((float) M_PI) / -s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))))) + -1.0f));
}

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log(((single(1.0) / ((u / (single(1.0) + exp((single(pi) / -s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))))) + single(-1.0)));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} \]
4. Add Preprocessing

5. Final simplification 98.8%

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

Alternative 2: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{2} + \frac{1 - u}{1 + e^{\frac{1}{\frac{s}{\pi}}}}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     -1.0
     (/ 1.0 (+ (/ u 2.0) (/ (- 1.0 u) (+ 1.0 (exp (/ 1.0 (/ s PI))))))))))))

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / ((u / 2.0f) + ((1.0f - u) / (1.0f + expf((1.0f / (s / ((float) M_PI))))))))));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u / Float32(2.0)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(1.0) / Float32(s / Float32(pi))))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / ((u / single(2.0)) + ((single(1.0) - u) / (single(1.0) + exp((single(1.0) / (s / single(pi))))))))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 37.8%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + \color{blue}{1}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right) \]
6. Step-by-step derivation

   1. clear-num 37.8%

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

   2. inv-pow 37.8%

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

7. Applied egg-rr 37.8%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + 1} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\frac{s}{\pi}\right)}^{-1}}}}} + -1\right)\right) \]
8. Step-by-step derivation

   1. unpow-1 37.8%

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

9. Applied egg-rr 37.8%

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

10. Final simplification 37.8%

    \[\leadsto s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{2} + \frac{1 - u}{1 + e^{\frac{1}{\frac{s}{\pi}}}}}\right)\right) \]
11. Add Preprocessing

Alternative 3: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(-1 + \frac{1}{\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + \frac{u}{2}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log (+ -1.0 (/ 1.0 (+ (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))) (/ u 2.0))))))))

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / (((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))) + (u / 2.0f)))));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) + Float32(u / Float32(2.0))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(1.0) / (((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))) + (u / single(2.0))))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 37.8%

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

6. Final simplification 37.8%

   \[\leadsto s \cdot \left(-\log \left(-1 + \frac{1}{\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + \frac{u}{2}}\right)\right) \]
7. Add Preprocessing

Alternative 4: 37.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(-1 + \frac{2}{u}\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s (- (log (+ -1.0 (/ 2.0 u))))))

C

float code(float u, float s) {
	return s * -logf((-1.0f + (2.0f / u)));
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * -log(((-1.0e0) + (2.0e0 / u)))
end function

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(-1.0) + Float32(Float32(2.0) / u)))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(2.0) / u)));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 37.8%

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

6. Taylor expanded in s around inf 36.4%

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

7. Taylor expanded in s around 0 37.3%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(2 \cdot \frac{1}{u} - 1\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* 37.3%

      \[\leadsto \color{blue}{\left(-1 \cdot s\right) \cdot \log \left(2 \cdot \frac{1}{u} - 1\right)} \]

   2. neg-mul-1 37.3%

      \[\leadsto \color{blue}{\left(-s\right)} \cdot \log \left(2 \cdot \frac{1}{u} - 1\right) \]

   3. sub-neg 37.3%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(2 \cdot \frac{1}{u} + \left(-1\right)\right)} \]

   4. associate-*r/ 37.3%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{2 \cdot 1}{u}} + \left(-1\right)\right) \]

   5. metadata-eval 37.3%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{\color{blue}{2}}{u} + \left(-1\right)\right) \]

   6. metadata-eval 37.3%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{2}{u} + \color{blue}{-1}\right) \]

9. Simplified 37.3%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{2}{u} + -1\right)} \]

10. Final simplification 37.3%

    \[\leadsto s \cdot \left(-\log \left(-1 + \frac{2}{u}\right)\right) \]
11. Add Preprocessing

Alternative 5: 25.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (/ PI s))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 37.8%

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

6. Taylor expanded in s around inf 36.4%

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

7. Taylor expanded in u around 0 25.3%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* 25.3%

      \[\leadsto \color{blue}{\left(-1 \cdot s\right) \cdot \log \left(1 + \frac{\pi}{s}\right)} \]

   2. neg-mul-1 25.3%

      \[\leadsto \color{blue}{\left(-s\right)} \cdot \log \left(1 + \frac{\pi}{s}\right) \]

   3. log1p-def 25.3%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Simplified 25.3%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Final simplification 25.3%

    \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
11. Add Preprocessing

Alternative 6: 11.8% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(\pi \cdot \left(u \cdot 0.5 + -0.25\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* 4.0 (* PI (+ (* u 0.5) -0.25))))

C

float code(float u, float s) {
	return 4.0f * (((float) M_PI) * ((u * 0.5f) + -0.25f));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(pi) * Float32(Float32(u * Float32(0.5)) + Float32(-0.25))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (single(pi) * ((u * single(0.5)) + single(-0.25)));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in s around inf 11.6%

   \[\leadsto \color{blue}{4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right)} \]
6. Step-by-step derivation

   1. associate--r+ 11.6%

      \[\leadsto 4 \cdot \color{blue}{\left(\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) - 0.25 \cdot \pi\right)} \]

   2. cancel-sign-sub-inv 11.6%

      \[\leadsto 4 \cdot \color{blue}{\left(\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) + \left(-0.25\right) \cdot \pi\right)} \]

   3. distribute-rgt-out-- 11.6%

      \[\leadsto 4 \cdot \left(\color{blue}{\left(u \cdot \pi\right) \cdot \left(0.25 - -0.25\right)} + \left(-0.25\right) \cdot \pi\right) \]

   4. *-commutative 11.6%

      \[\leadsto 4 \cdot \left(\color{blue}{\left(\pi \cdot u\right)} \cdot \left(0.25 - -0.25\right) + \left(-0.25\right) \cdot \pi\right) \]

   5. metadata-eval 11.6%

      \[\leadsto 4 \cdot \left(\left(\pi \cdot u\right) \cdot \color{blue}{0.5} + \left(-0.25\right) \cdot \pi\right) \]

   6. metadata-eval 11.6%

      \[\leadsto 4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{-0.25} \cdot \pi\right) \]

   7. *-commutative 11.6%

      \[\leadsto 4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\pi \cdot -0.25}\right) \]

7. Simplified 11.6%

   \[\leadsto \color{blue}{4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25\right)} \]

8. Taylor expanded in u around 0 11.6%

   \[\leadsto 4 \cdot \color{blue}{\left(-0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative 11.6%

      \[\leadsto 4 \cdot \color{blue}{\left(0.5 \cdot \left(u \cdot \pi\right) + -0.25 \cdot \pi\right)} \]

   2. associate-*r* 11.6%

      \[\leadsto 4 \cdot \left(\color{blue}{\left(0.5 \cdot u\right) \cdot \pi} + -0.25 \cdot \pi\right) \]

   3. distribute-rgt-out 11.6%

      \[\leadsto 4 \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot u + -0.25\right)\right)} \]

10. Simplified 11.6%

    \[\leadsto 4 \cdot \color{blue}{\left(\pi \cdot \left(0.5 \cdot u + -0.25\right)\right)} \]

11. Final simplification 11.6%

    \[\leadsto 4 \cdot \left(\pi \cdot \left(u \cdot 0.5 + -0.25\right)\right) \]
12. Add Preprocessing

Alternative 7: 11.5% accurate, 216.5× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- PI))

C

float code(float u, float s) {
	return -((float) M_PI);
}

Julia

function code(u, s)
	return Float32(-Float32(pi))
end

MATLAB

function tmp = code(u, s)
	tmp = -single(pi);
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in u around 0 11.6%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
6. Step-by-step derivation

   1. neg-mul-1 11.6%

      \[\leadsto \color{blue}{-\pi} \]

7. Simplified 11.6%

   \[\leadsto \color{blue}{-\pi} \]

8. Final simplification 11.6%

   \[\leadsto -\pi \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))))