Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 10.7s

Alternatives: 6

Speedup: 1.4×

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: 98.9% 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: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 25.1% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around -inf 25.6%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)}\right) \]
5. Step-by-step derivation

   1. +-commutative 25.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s} + 1\right)}\right) \]

   2. fma-def 25.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)}\right) \]

6. Simplified 25.6%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)}\right) \]

7. Taylor expanded in s around 0 25.8%

   \[\leadsto s \cdot \left(-\color{blue}{\left(\log \left(4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) + \pi \cdot \left(0.25 + -0.25 \cdot u\right)\right)\right) + -1 \cdot \log s\right)}\right) \]

8. Taylor expanded in u around 0 25.9%

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

9. Taylor expanded in u around inf 26.0%

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

10. Final simplification 26.0%

    \[\leadsto s \cdot \left(\log s - u \cdot -2\right) \]

Alternative 3: 11.5% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around -inf 25.6%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)}\right) \]
5. Step-by-step derivation

   1. +-commutative 25.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s} + 1\right)}\right) \]

   2. fma-def 25.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)}\right) \]

6. Simplified 25.6%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)}\right) \]
7. Step-by-step derivation

   1. distribute-rgt-neg-out 25.6%

      \[\leadsto \color{blue}{-s \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \]

   2. neg-sub0 25.6%

      \[\leadsto \color{blue}{0 - s \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \]

   3. add-sqr-sqrt 25.6%

      \[\leadsto 0 - s \cdot \color{blue}{\left(\sqrt{\log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \cdot \sqrt{\log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)}\right)} \]

   4. sqrt-unprod 25.6%

      \[\leadsto 0 - s \cdot \color{blue}{\sqrt{\log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)}} \]

   5. sqr-neg 25.6%

      \[\leadsto 0 - s \cdot \sqrt{\color{blue}{\left(-\log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)\right) \cdot \left(-\log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)\right)}} \]

   6. sqrt-unprod -0.0%

      \[\leadsto 0 - s \cdot \color{blue}{\left(\sqrt{-\log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)} \cdot \sqrt{-\log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)}\right)} \]

8. Applied egg-rr 25.6%

   \[\leadsto \color{blue}{0 - s \cdot \mathsf{log1p}\left(4 \cdot \frac{\mathsf{fma}\left(\pi, \mathsf{fma}\left(u, -0.25, 0.25\right), \pi \cdot \left(u \cdot -0.25\right)\right)}{s}\right)} \]
9. Step-by-step derivation

   1. neg-sub0 25.6%

      \[\leadsto \color{blue}{-s \cdot \mathsf{log1p}\left(4 \cdot \frac{\mathsf{fma}\left(\pi, \mathsf{fma}\left(u, -0.25, 0.25\right), \pi \cdot \left(u \cdot -0.25\right)\right)}{s}\right)} \]

   2. *-commutative 25.6%

      \[\leadsto -\color{blue}{\mathsf{log1p}\left(4 \cdot \frac{\mathsf{fma}\left(\pi, \mathsf{fma}\left(u, -0.25, 0.25\right), \pi \cdot \left(u \cdot -0.25\right)\right)}{s}\right) \cdot s} \]

   3. distribute-rgt-neg-in 25.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(4 \cdot \frac{\mathsf{fma}\left(\pi, \mathsf{fma}\left(u, -0.25, 0.25\right), \pi \cdot \left(u \cdot -0.25\right)\right)}{s}\right) \cdot \left(-s\right)} \]

10. Simplified 25.6%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(4 \cdot \frac{\pi \cdot \left(u \cdot -0.25 + \mathsf{fma}\left(u, -0.25, 0.25\right)\right)}{s}\right) \cdot \left(-s\right)} \]

11. Taylor expanded in s around inf 11.9%

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

    1. associate-*r* 11.9%

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

    2. *-commutative 11.9%

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

13. Simplified 11.9%

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

14. Taylor expanded in u around 0 11.9%

    \[\leadsto \color{blue}{-1 \cdot \pi + 2 \cdot \left(u \cdot \pi\right)} \]
15. Step-by-step derivation

    1. +-commutative 11.9%

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

    2. associate-*r* 11.9%

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

    3. distribute-rgt-out 11.9%

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

    4. *-commutative 11.9%

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

16. Simplified 11.9%

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

17. Final simplification 11.9%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]

Alternative 4: 11.3% accurate, 7.2× 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 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u around 0 11.8%

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

   1. neg-mul-1 11.8%

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

6. Simplified 11.8%

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

7. Final simplification 11.8%

   \[\leadsto -\pi \]

Alternative 5: 9.0% accurate, 146.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(s \cdot u\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* 2.0 (* s u)))

C

float code(float u, float s) {
	return 2.0f * (s * u);
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 2.0e0 * (s * u)
end function

Julia

function code(u, s)
	return Float32(Float32(2.0) * Float32(s * u))
end

MATLAB

function tmp = code(u, s)
	tmp = single(2.0) * (s * u);
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around -inf 25.6%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)}\right) \]
5. Step-by-step derivation

   1. +-commutative 25.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s} + 1\right)}\right) \]

   2. fma-def 25.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)}\right) \]

6. Simplified 25.6%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)}\right) \]

7. Taylor expanded in s around 0 25.8%

   \[\leadsto s \cdot \left(-\color{blue}{\left(\log \left(4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) + \pi \cdot \left(0.25 + -0.25 \cdot u\right)\right)\right) + -1 \cdot \log s\right)}\right) \]

8. Taylor expanded in u around 0 25.9%

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

9. Taylor expanded in u around inf 8.4%

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

10. Final simplification 8.4%

    \[\leadsto 2 \cdot \left(s \cdot u\right) \]

Alternative 6: 9.0% accurate, 146.0× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 2\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* u (* s 2.0)))

C

float code(float u, float s) {
	return u * (s * 2.0f);
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = u * (s * 2.0e0)
end function

Julia

function code(u, s)
	return Float32(u * Float32(s * Float32(2.0)))
end

MATLAB

function tmp = code(u, s)
	tmp = u * (s * single(2.0));
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around -inf 25.6%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)}\right) \]
5. Step-by-step derivation

   1. +-commutative 25.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s} + 1\right)}\right) \]

   2. fma-def 25.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}, 1\right)\right)}\right) \]

6. Simplified 25.6%

   \[\leadsto s \cdot \left(-\log \color{blue}{\left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + u \cdot \left(\pi \cdot -0.25\right)}{s}, 1\right)\right)}\right) \]

7. Taylor expanded in s around 0 25.8%

   \[\leadsto s \cdot \left(-\color{blue}{\left(\log \left(4 \cdot \left(-0.25 \cdot \left(u \cdot \pi\right) + \pi \cdot \left(0.25 + -0.25 \cdot u\right)\right)\right) + -1 \cdot \log s\right)}\right) \]

8. Taylor expanded in u around 0 25.9%

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

9. Taylor expanded in u around inf 8.4%

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

    1. associate-*r* 8.4%

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

    2. *-commutative 8.4%

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

    3. *-commutative 8.4%

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

11. Simplified 8.4%

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

12. Final simplification 8.4%

    \[\leadsto u \cdot \left(s \cdot 2\right) \]

Reproduce

herbie shell --seed 2023306 
(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))))