Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 12.1s

Alternatives: 9

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, 1.2× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 1.0 (pow E (/ 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 + powf(((float) M_E), (((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) + (Float32(exp(1)) ^ 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) + (single(2.71828182845904523536) ^ (single(pi) / s)))))) + single(-1.0)));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. exp-prod 99.0%

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

   3. exp-1-e 99.0%

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

5. Applied egg-rr 99.0%

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

6. Final simplification 99.0%

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

Alternative 2: 98.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     -1.0
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 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 / (1.0f + expf((((float) M_PI) / -s)))) + ((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(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + 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(1.0) + exp((single(pi) / -s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(1.0) / (s / single(pi))))))))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 99.0%

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

   3. sqr-neg 99.0%

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

   4. sqrt-unprod -0.0%

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

   5. add-cbrt-cube -0.0%

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

   6. add-sqr-sqrt 2.3%

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

   7. add-cbrt-cube 2.3%

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

   8. cbrt-undiv 2.3%

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

   9. pow3 2.3%

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

   10. pow3 2.3%

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

   11. add-sqr-sqrt -0.0%

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

   12. sqrt-unprod 99.0%

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

   13. sqr-neg 99.0%

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

   14. sqrt-unprod 99.0%

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

   15. add-sqr-sqrt 99.0%

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

5. Applied egg-rr 99.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\sqrt[3]{\frac{{\pi}^{3}}{{s}^{3}}}}}}} + -1\right) \]
6. Step-by-step derivation

   1. cbrt-div 99.0%

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

   2. unpow3 99.0%

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

   3. add-cbrt-cube 99.0%

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

   4. unpow3 99.0%

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

   5. add-cbrt-cube 99.0%

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

   6. clear-num 99.0%

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Alternative 3: 98.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     -1.0
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 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)))) + ((1.0f - u) / (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(Float32(1.0) - u) / 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)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s))))))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 4: 86.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / ((u / (1.0f + expf((((float) M_PI) / -s)))) + ((1.0f - u) / (1.0f + (1.0f + (((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(Float32(1.0) - u) / Float32(Float32(1.0) + Float32(Float32(1.0) + 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)))) + ((single(1.0) - u) / (single(1.0) + (single(1.0) + (single(pi) / s))))))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 87.8%

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

   1. +-commutative 87.8%

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

6. Simplified 87.8%

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

7. Final simplification 87.8%

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

Alternative 5: 37.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{\frac{-1}{u}}{0.5 + \frac{-1}{1 + e^{\frac{-\pi}{s}}}}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log (+ -1.0 (/ (/ -1.0 u) (+ 0.5 (/ -1.0 (+ 1.0 (exp (/ (- PI) s))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + ((-1.0f / u) / (0.5f + (-1.0f / (1.0f + expf((-((float) M_PI) / s))))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(Float32(-1.0) / u) / Float32(Float32(0.5) + Float32(Float32(-1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + ((single(-1.0) / u) / (single(0.5) + (single(-1.0) / (single(1.0) + exp((-single(pi) / s))))))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 9.5%

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

5. Taylor expanded in u around -inf 36.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{-1}{u \cdot \left(0.5 - \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}}\right)}} + -1\right) \]
6. Step-by-step derivation

   1. associate-/r* 36.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{\frac{-1}{u}}{0.5 - \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}}}} + -1\right) \]

   2. associate-*r/ 36.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{-1}{u}}{0.5 - \frac{1}{1 + e^{\color{blue}{\frac{-1 \cdot \pi}{s}}}}} + -1\right) \]

   3. neg-mul-1 36.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{-1}{u}}{0.5 - \frac{1}{1 + e^{\frac{\color{blue}{-\pi}}{s}}}} + -1\right) \]

7. Simplified 36.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{\frac{-1}{u}}{0.5 - \frac{1}{1 + e^{\frac{-\pi}{s}}}}} + -1\right) \]

8. Final simplification 36.9%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{\frac{-1}{u}}{0.5 + \frac{-1}{1 + e^{\frac{-\pi}{s}}}}\right) \]

Alternative 6: 12.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \sqrt[3]{-0.125 \cdot {\left(u \cdot \pi\right)}^{3}} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* 4.0 (cbrt (* -0.125 (pow (* u PI) 3.0)))))

C

float code(float u, float s) {
	return 4.0f * cbrtf((-0.125f * powf((u * ((float) M_PI)), 3.0f)));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * cbrt(Float32(Float32(-0.125) * (Float32(u * Float32(pi)) ^ Float32(3.0)))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 11.8%

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

5. Taylor expanded in u around inf 5.1%

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

   1. distribute-rgt-out-- 5.1%

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

   2. metadata-eval 5.1%

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

   3. *-commutative 5.1%

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

   4. *-commutative 5.1%

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

   5. *-commutative 5.1%

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

   6. associate-*l* 5.1%

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

7. Simplified 5.1%

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

   1. *-commutative 5.1%

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

   2. add-sqr-sqrt 5.1%

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

   3. sqrt-unprod 5.1%

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

   4. swap-sqr 5.1%

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

   5. metadata-eval 5.1%

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

   6. metadata-eval 5.1%

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

   7. swap-sqr 5.1%

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

   8. *-commutative 5.1%

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

   9. *-commutative 5.1%

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

   10. sqrt-unprod -0.0%

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

   11. add-sqr-sqrt 11.9%

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

   12. add-cbrt-cube 11.9%

       \[\leadsto 4 \cdot \left(\color{blue}{\sqrt[3]{\left(\left(u \cdot -0.5\right) \cdot \left(u \cdot -0.5\right)\right) \cdot \left(u \cdot -0.5\right)}} \cdot \pi\right) \]

   13. add-cbrt-cube 11.9%

       \[\leadsto 4 \cdot \left(\sqrt[3]{\left(\left(u \cdot -0.5\right) \cdot \left(u \cdot -0.5\right)\right) \cdot \left(u \cdot -0.5\right)} \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \]

   14. cbrt-unprod 11.9%

       \[\leadsto 4 \cdot \color{blue}{\sqrt[3]{\left(\left(\left(u \cdot -0.5\right) \cdot \left(u \cdot -0.5\right)\right) \cdot \left(u \cdot -0.5\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}} \]

   15. pow3 11.9%

       \[\leadsto 4 \cdot \sqrt[3]{\color{blue}{{\left(u \cdot -0.5\right)}^{3}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)} \]

   16. unpow-prod-down 11.9%

       \[\leadsto 4 \cdot \sqrt[3]{\color{blue}{\left({u}^{3} \cdot {-0.5}^{3}\right)} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)} \]

   17. metadata-eval 11.9%

       \[\leadsto 4 \cdot \sqrt[3]{\left({u}^{3} \cdot \color{blue}{-0.125}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)} \]

   18. pow3 11.9%

       \[\leadsto 4 \cdot \sqrt[3]{\left({u}^{3} \cdot -0.125\right) \cdot \color{blue}{{\pi}^{3}}} \]

9. Applied egg-rr 11.9%

   \[\leadsto 4 \cdot \color{blue}{\sqrt[3]{\left({u}^{3} \cdot -0.125\right) \cdot {\pi}^{3}}} \]
10. Step-by-step derivation

    1. *-commutative 11.9%

       \[\leadsto 4 \cdot \sqrt[3]{\color{blue}{\left(-0.125 \cdot {u}^{3}\right)} \cdot {\pi}^{3}} \]

    2. associate-*l* 11.9%

       \[\leadsto 4 \cdot \sqrt[3]{\color{blue}{-0.125 \cdot \left({u}^{3} \cdot {\pi}^{3}\right)}} \]

    3. cube-prod 11.9%

       \[\leadsto 4 \cdot \sqrt[3]{-0.125 \cdot \color{blue}{{\left(u \cdot \pi\right)}^{3}}} \]

11. Simplified 11.9%

    \[\leadsto 4 \cdot \color{blue}{\sqrt[3]{-0.125 \cdot {\left(u \cdot \pi\right)}^{3}}} \]

12. Final simplification 11.9%

    \[\leadsto 4 \cdot \sqrt[3]{-0.125 \cdot {\left(u \cdot \pi\right)}^{3}} \]

Alternative 7: 11.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. associate-*r* 11.8%

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

   4. distribute-rgt-out-- 11.8%

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

   5. *-commutative 11.8%

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

   6. metadata-eval 11.8%

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

   7. *-commutative 11.8%

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

   8. *-commutative 11.8%

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

   9. associate-*l* 11.8%

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

6. Simplified 11.8%

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

7. Final simplification 11.8%

   \[\leadsto -4 \cdot \left(\pi \cdot \left(u \cdot -0.25 - -0.25\right) + \pi \cdot \left(u \cdot -0.25\right)\right) \]

Alternative 8: 11.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. associate-*r* 11.8%

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

   4. distribute-rgt-out-- 11.8%

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

   5. *-commutative 11.8%

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

   6. metadata-eval 11.8%

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

   7. *-commutative 11.8%

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

   8. *-commutative 11.8%

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

   9. associate-*l* 11.8%

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

6. Simplified 11.8%

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

   1. distribute-lft-out 11.8%

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

   2. fma-neg 11.8%

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

   3. metadata-eval 11.8%

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

8. Applied egg-rr 11.8%

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

9. Taylor expanded in u around 0 11.8%

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

    1. *-commutative 11.8%

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

11. Simplified 11.8%

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

12. Final simplification 11.8%

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

Alternative 9: 11.2% 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 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u around 0 11.6%

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

   1. neg-mul-1 11.6%

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

6. Simplified 11.6%

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

7. Final simplification 11.6%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023271 
(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))))