Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 26.8s

Alternatives: 9

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: 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 98.9%

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

   1. expm1-log1p-u 96.6%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)\right)}\right) \]

5. Applied egg-rr 96.6%

   \[\leadsto s \cdot \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. expm1-udef 96.5%

      \[\leadsto s \cdot \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} - 1\right)}\right) \]

   2. log1p-udef 96.6%

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

   3. add-exp-log 98.8%

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

7. Applied egg-rr 98.8%

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

   1. associate--l+ 98.9%

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

9. Simplified 98.9%

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

    1. add-log-exp 99.0%

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

    2. sub-neg 99.0%

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

11. Applied egg-rr 99.0%

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

12. Final simplification 99.0%

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

Alternative 2: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

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 / (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(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) + Float32(u / 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) / (((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))) + (u / (single(1.0) + exp((-single(pi) / s))))))));
end

Derivation

1. Initial program 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 3: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\mathsf{log1p}\left(\frac{1}{\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{\frac{-\pi}{s}}}} + -2\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log1p
    (+
     (/
      1.0
      (+
       (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))
       (/ u (+ 1.0 (exp (/ (- PI) s))))))
     -2.0)))))

C

float code(float u, float s) {
	return s * -log1pf(((1.0f / (((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))) + (u / (1.0f + expf((-((float) M_PI) / s)))))) + -2.0f));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log1p(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) + Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))))) + Float32(-2.0)))))
end

Derivation

1. Initial program 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 98.9%

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

   1. log1p-expm1-u 98.9%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)\right)}\right) \]

   2. expm1-udef 98.9%

      \[\leadsto s \cdot \left(-\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} - 1}\right)\right) \]

   3. add-exp-log 99.0%

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

5. Applied egg-rr 99.0%

   \[\leadsto s \cdot \left(-\color{blue}{\mathsf{log1p}\left(\left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right) - 1\right)}\right) \]
6. Step-by-step derivation

   1. associate--l+ 99.0%

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

   2. metadata-eval 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

   \[\leadsto s \cdot \left(-\mathsf{log1p}\left(\frac{1}{\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + \frac{u}{1 + e^{\frac{-\pi}{s}}}} + -2\right)\right) \]

Alternative 4: 23.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\log \left(1 + -2 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(\left(u \cdot \pi\right) \cdot -0.25 + \pi \cdot 0.25\right)}{s}\right) \cdot \left(-2\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (*
   (log
    (+
     1.0
     (* -2.0 (/ (- (* 0.25 (* u PI)) (+ (* (* u PI) -0.25) (* PI 0.25))) s))))
   (- 2.0))))

C

float code(float u, float s) {
	return s * (logf((1.0f + (-2.0f * (((0.25f * (u * ((float) M_PI))) - (((u * ((float) M_PI)) * -0.25f) + (((float) M_PI) * 0.25f))) / s)))) * -2.0f);
}

Julia

function code(u, s)
	return Float32(s * Float32(log(Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(Float32(Float32(0.25) * Float32(u * Float32(pi))) - Float32(Float32(Float32(u * Float32(pi)) * Float32(-0.25)) + Float32(Float32(pi) * Float32(0.25)))) / s)))) * Float32(-Float32(2.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * (log((single(1.0) + (single(-2.0) * (((single(0.25) * (u * single(pi))) - (((u * single(pi)) * single(-0.25)) + (single(pi) * single(0.25)))) / s)))) * -single(2.0));
end

Derivation

1. Initial program 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 98.9%

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

   1. add-sqr-sqrt 98.9%

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

   2. log-prod 98.9%

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

5. Applied egg-rr 98.9%

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

   1. count-2 98.9%

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

   2. +-commutative 98.9%

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

7. Simplified 98.9%

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

8. Taylor expanded in s around inf 23.0%

   \[\leadsto s \cdot \left(-2 \cdot \log \color{blue}{\left(-2 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s} + 1\right)}\right) \]

9. Final simplification 23.0%

   \[\leadsto s \cdot \left(\log \left(1 + -2 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(\left(u \cdot \pi\right) \cdot -0.25 + \pi \cdot 0.25\right)}{s}\right) \cdot \left(-2\right)\right) \]

Alternative 5: 11.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 98.9%

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

4. Taylor expanded in s around inf 11.3%

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

5. Final simplification 11.3%

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

Alternative 6: 11.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 98.9%

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

   1. expm1-log1p-u 96.6%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)\right)}\right) \]

   2. expm1-udef 96.5%

      \[\leadsto s \cdot \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} - 1\right)}\right) \]

5. Applied egg-rr 96.5%

   \[\leadsto s \cdot \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} - 1\right)}\right) \]

6. Taylor expanded in s around inf 11.3%

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

7. Taylor expanded in u around 0 11.2%

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

   1. fma-def 11.2%

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

   2. distribute-rgt-out-- 11.2%

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

   3. metadata-eval 11.2%

      \[\leadsto s \cdot \left(-\left(\left(-4 \cdot \mathsf{fma}\left(\frac{\pi}{s} \cdot \color{blue}{0.5}, u, -0.25 \cdot \frac{\pi}{s}\right) + 1\right) - 1\right)\right) \]

   4. *-commutative 11.2%

      \[\leadsto s \cdot \left(-\left(\left(-4 \cdot \mathsf{fma}\left(\frac{\pi}{s} \cdot 0.5, u, \color{blue}{\frac{\pi}{s} \cdot -0.25}\right) + 1\right) - 1\right)\right) \]

9. Simplified 11.2%

   \[\leadsto s \cdot \left(-\left(\left(-4 \cdot \color{blue}{\mathsf{fma}\left(\frac{\pi}{s} \cdot 0.5, u, \frac{\pi}{s} \cdot -0.25\right)} + 1\right) - 1\right)\right) \]

10. Taylor expanded in u around 0 11.3%

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

    1. mul-1-neg 11.3%

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

    2. unsub-neg 11.3%

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

    3. *-commutative 11.3%

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

    4. *-commutative 11.3%

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

12. Simplified 11.3%

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

13. Final simplification 11.3%

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

Alternative 7: 11.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. sub-neg 99.0%

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

3. Simplified 99.0%

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

   1. expm1-log1p-u 97.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(u, -1, 1\right), \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{u}{1 + e^{\frac{-\pi}{s}}}\right)} + -1\right)\right)\right)} \]

   2. expm1-udef 20.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(u, -1, 1\right), \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{u}{1 + e^{\frac{-\pi}{s}}}\right)} + -1\right)\right)} - 1} \]

   3. +-commutative 20.9%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \color{blue}{\left(-1 + \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(u, -1, 1\right), \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{u}{1 + e^{\frac{-\pi}{s}}}\right)}\right)}\right)} - 1 \]

5. Applied egg-rr 20.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(-1 + \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(u, -1, 1\right), \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{u}{1 + e^{\frac{-\pi}{s}}}\right)}\right)\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 97.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(-1 + \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(u, -1, 1\right), \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{u}{1 + e^{\frac{-\pi}{s}}}\right)}\right)\right)\right)} \]

   2. expm1-log1p 99.0%

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

   3. fma-udef 99.0%

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

   4. *-commutative 99.0%

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

   5. +-commutative 99.0%

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

   6. mul-1-neg 99.0%

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

   7. sub-neg 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in u around 0 11.0%

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

   1. associate-*r/ 11.0%

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

10. Applied egg-rr 11.0%

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

    1. associate-/l* 11.0%

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

12. Simplified 11.0%

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

13. Final simplification 11.0%

    \[\leadsto \frac{-s}{\frac{s}{\pi}} \]

Alternative 8: 11.4% 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%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 98.9%

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

4. Taylor expanded in u around 0 11.0%

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

   1. neg-mul-1 11.0%

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

6. Simplified 11.0%

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

7. Final simplification 11.0%

   \[\leadsto -\pi \]

Alternative 9: 10.3% accurate, 243.3× speedup

Math

\[\begin{array}{l} \\ s \cdot 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s 0.0))

C

float code(float u, float s) {
	return s * 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * 0.0e0
end function

Julia

function code(u, s)
	return Float32(s * Float32(0.0))
end

MATLAB

function tmp = code(u, s)
	tmp = s * single(0.0);
end

Derivation

1. Initial program 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 98.9%

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

   1. expm1-log1p-u 96.6%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)\right)}\right) \]

   2. expm1-udef 96.5%

      \[\leadsto s \cdot \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} - 1\right)}\right) \]

5. Applied egg-rr 96.5%

   \[\leadsto s \cdot \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} - 1\right)}\right) \]

6. Taylor expanded in s around inf 10.3%

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

7. Final simplification 10.3%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023228 
(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))))