Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 16.8s

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

\[\begin{array}{l} \\ s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{1}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{s}{\pi}\right)\right)}}}} + -1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ (- PI) s))))
       (/ (- 1.0 u) (+ 1.0 (exp (/ 1.0 (log1p (expm1 (/ s PI)))))))))
     -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((1.0f / log1pf(expm1f((s / ((float) M_PI)))))))))) + -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(-Float32(pi)) / s)))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(1.0) / log1p(expm1(Float32(s / Float32(pi)))))))))) + Float32(-1.0)))))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 99.0%

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

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

5. Applied egg-rr 99.0%

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

   1. unpow-1 99.0%

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

7. Simplified 99.0%

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

   1. log1p-expm1-u 99.0%

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

9. Applied egg-rr 99.0%

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

10. Final simplification 99.0%

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

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(-Float32(pi)) / 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%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 99.0%

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

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

5. Applied egg-rr 99.0%

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

   1. unpow-1 99.0%

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

7. Simplified 99.0%

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

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(-Float32(pi)) / 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%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 99.0%

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

4. Final simplification 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: 37.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(-\log \left(-1 + \frac{1}{\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + u \cdot 0.5}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (log1p
  (expm1
   (*
    s
    (-
     (log
      (+ -1.0 (/ 1.0 (+ (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))) (* u 0.5))))))))))

C

float code(float u, float s) {
	return log1pf(expm1f((s * -logf((-1.0f + (1.0f / (((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))) + (u * 0.5f))))))));
}

Julia

function code(u, s)
	return log1p(expm1(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(0.5))))))))))
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 99.0%

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

4. Taylor expanded in s around inf 38.0%

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

   1. log1p-expm1-u 38.0%

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

   2. +-commutative 38.0%

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

   3. div-inv 38.0%

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

   4. metadata-eval 38.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(-\log \left(-1 + \frac{1}{u \cdot \frac{1}{\color{blue}{2}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right)\right)\right) \]

   5. metadata-eval 38.0%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(-\log \left(-1 + \frac{1}{u \cdot \color{blue}{0.5} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right)\right)\right) \]

6. Applied egg-rr 38.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(-\log \left(-1 + \frac{1}{u \cdot 0.5 + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right)\right)\right)} \]

7. Final simplification 38.0%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(-\log \left(-1 + \frac{1}{\frac{1 - u}{1 + e^{\frac{\pi}{s}}} + u \cdot 0.5}\right)\right)\right)\right) \]

Alternative 5: 37.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = log((single(-1.0) + (single(1.0) / (((single(1.0) - u) / (single(1.0) + exp((single(1.0) / (s / single(pi)))))) + (u * single(0.5)))))) * -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 99.0%

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

4. Taylor expanded in s around inf 38.0%

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

   1. distribute-rgt-neg-out 38.0%

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

   2. +-commutative 38.0%

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

   3. div-inv 38.0%

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

   4. metadata-eval 38.0%

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

   5. metadata-eval 38.0%

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

6. Applied egg-rr 38.0%

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

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

8. Applied egg-rr 38.0%

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

   1. unpow-1 99.0%

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

10. Simplified 38.0%

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

11. Final simplification 38.0%

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

Alternative 6: 37.6% accurate, 2.3× speedup

Math

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

FPCore

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

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 * 0.5f)))));
}

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(0.5))))))))
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(0.5))))));
end

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 99.0%

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

4. Taylor expanded in s around inf 38.0%

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

   1. distribute-rgt-neg-out 38.0%

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

   2. +-commutative 38.0%

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

   3. div-inv 38.0%

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

   4. metadata-eval 38.0%

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

   5. metadata-eval 38.0%

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

6. Applied egg-rr 38.0%

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

7. Final simplification 38.0%

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

Alternative 7: 11.4% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 99.0%

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

4. Taylor expanded in s around inf 38.0%

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

5. Taylor expanded in s around inf 11.8%

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

   1. cancel-sign-sub-inv 11.8%

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

   2. associate-*r* 11.8%

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

   3. metadata-eval 11.8%

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

   4. distribute-rgt-out 11.8%

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

7. Simplified 11.8%

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

8. Final simplification 11.8%

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

Alternative 8: 37.0% accurate, 6.8× speedup

Math

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

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(Float32(-s) * 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 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 99.0%

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

4. Taylor expanded in s around inf 38.0%

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

5. Taylor expanded in s around inf 36.2%

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

   1. +-commutative 36.2%

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

7. Simplified 36.2%

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

8. Taylor expanded in s around 0 37.2%

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

   1. associate-*r* 37.2%

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

   2. neg-mul-1 37.2%

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

   3. sub-neg 37.2%

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

   4. associate-*r/ 37.2%

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

   5. metadata-eval 37.2%

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

   6. metadata-eval 37.2%

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

10. Simplified 37.2%

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

11. Final simplification 37.2%

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

Alternative 9: 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 99.0%

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

4. Taylor expanded in u around 0 11.8%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
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%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023258 
(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))))