Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 15.2s

Alternatives: 11

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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. Add Preprocessing

5. Final simplification 99.0%

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

Alternative 2: 25.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * (((u * single(2.0)) - (s / single(pi))) + (log(s) - log(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. Add Preprocessing

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.2%

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

   1. mul-1-neg 25.2%

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

   2. unsub-neg 25.2%

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

8. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{-4 \cdot \frac{\left(u \cdot s\right) \cdot \left(\pi \cdot -0.5\right)}{s}}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Taylor expanded in s around 0 25.2%

   \[\leadsto \color{blue}{2 \cdot \left(s \cdot u\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
10. Step-by-step derivation

    1. *-commutative 25.2%

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

11. Simplified 25.2%

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

12. Taylor expanded in s around 0 25.3%

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

    1. +-commutative 25.3%

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

    2. mul-1-neg 25.3%

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

    3. unsub-neg 25.3%

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

    4. mul-1-neg 25.3%

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

    5. unsub-neg 25.3%

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

14. Simplified 25.3%

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

15. Final simplification 25.3%

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

Alternative 3: 25.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * ((u * single(2.0)) + (log(s) - log(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. Add Preprocessing

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.2%

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

   1. mul-1-neg 25.2%

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

   2. unsub-neg 25.2%

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

8. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{-4 \cdot \frac{\left(u \cdot s\right) \cdot \left(\pi \cdot -0.5\right)}{s}}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Taylor expanded in s around 0 25.2%

   \[\leadsto \color{blue}{2 \cdot \left(s \cdot u\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
10. Step-by-step derivation

    1. *-commutative 25.2%

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

11. Simplified 25.2%

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

12. Taylor expanded in s around 0 25.3%

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

    1. mul-1-neg 25.3%

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

    2. unsub-neg 25.3%

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

14. Simplified 25.3%

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

15. Final simplification 25.3%

    \[\leadsto s \cdot \left(u \cdot 2 + \left(\log s - \log \pi\right)\right) \]
16. Add Preprocessing

Alternative 4: 25.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

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. Add Preprocessing

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.2%

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

   1. mul-1-neg 25.2%

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

   2. unsub-neg 25.2%

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

8. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{-4 \cdot \frac{\left(u \cdot s\right) \cdot \left(\pi \cdot -0.5\right)}{s}}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Taylor expanded in s around 0 25.2%

   \[\leadsto \color{blue}{2 \cdot \left(s \cdot u\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
10. Step-by-step derivation

    1. *-commutative 25.2%

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

11. Simplified 25.2%

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

12. Taylor expanded in u around inf 25.2%

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

13. Final simplification 25.2%

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

Alternative 5: 25.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

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. Add Preprocessing

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.2%

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

   1. mul-1-neg 25.2%

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

   2. unsub-neg 25.2%

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

8. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{-4 \cdot \frac{\left(u \cdot s\right) \cdot \left(\pi \cdot -0.5\right)}{s}}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Taylor expanded in s around 0 25.2%

   \[\leadsto \color{blue}{2 \cdot \left(s \cdot u\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
10. Step-by-step derivation

    1. *-commutative 25.2%

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

11. Simplified 25.2%

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

12. Taylor expanded in u around -inf 25.2%

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

    1. mul-1-neg 25.2%

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

    2. distribute-rgt-neg-in 25.2%

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

    3. +-commutative 25.2%

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

    4. log1p-define 25.2%

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

    5. associate-/l* 25.2%

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

    6. *-commutative 25.2%

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

    7. distribute-lft-out 25.2%

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

14. Simplified 25.2%

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

15. Final simplification 25.2%

    \[\leadsto u \cdot \left(\left(-s\right) \cdot \left(\frac{\mathsf{log1p}\left(\frac{\pi}{s}\right)}{u} + -2\right)\right) \]
16. Add Preprocessing

Alternative 6: 25.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(Float32(u * Float32(2.0)) - log1p(Float32(Float32(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. Add Preprocessing

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.2%

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

   1. mul-1-neg 25.2%

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

   2. unsub-neg 25.2%

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

8. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{-4 \cdot \frac{\left(u \cdot s\right) \cdot \left(\pi \cdot -0.5\right)}{s}}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Taylor expanded in s around 0 25.2%

   \[\leadsto \color{blue}{2 \cdot \left(s \cdot u\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
10. Step-by-step derivation

    1. *-commutative 25.2%

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

11. Simplified 25.2%

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

12. Taylor expanded in u around 0 25.2%

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

    1. *-commutative 25.2%

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

    2. associate-*l* 25.2%

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

    3. *-commutative 25.2%

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

    4. log1p-define 25.2%

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

    5. distribute-lft-out-- 25.2%

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

14. Simplified 25.2%

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

15. Final simplification 25.2%

    \[\leadsto s \cdot \left(u \cdot 2 - \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]
16. Add Preprocessing

Alternative 7: 25.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(-log1p(Float32(Float32(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. Add Preprocessing

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.2%

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

   1. mul-1-neg 25.2%

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

   2. distribute-rgt-neg-in 25.2%

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

   3. log1p-define 25.2%

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

8. Simplified 25.2%

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

9. Final simplification 25.2%

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

Alternative 8: 11.5% accurate, 33.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

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. Add Preprocessing

5. Taylor expanded in s around inf 11.8%

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

6. Taylor expanded in u around inf 11.8%

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

7. Taylor expanded in u around 0 11.8%

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

   1. distribute-rgt-out-- 11.8%

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

   2. metadata-eval 11.8%

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

   3. *-commutative 11.8%

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

   4. associate-*l* 11.8%

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

   5. distribute-rgt-in 11.8%

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

   6. *-commutative 11.8%

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

9. Simplified 11.8%

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

10. Final simplification 11.8%

    \[\leadsto 4 \cdot \left(u \cdot \frac{\pi \cdot \left(-0.25 + u \cdot 0.5\right)}{u}\right) \]
11. Add Preprocessing

Alternative 9: 11.5% accurate, 48.1× 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. Add Preprocessing

5. Taylor expanded in s around inf 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. distribute-rgt-out-- 11.8%

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

   4. *-commutative 11.8%

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

   5. metadata-eval 11.8%

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

   6. metadata-eval 11.8%

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

   7. *-commutative 11.8%

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

7. Simplified 11.8%

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

8. Taylor expanded in u around 0 11.8%

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

   1. associate-*r* 11.8%

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

   2. *-commutative 11.8%

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

   3. distribute-rgt-out 11.8%

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

10. Simplified 11.8%

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

11. Final simplification 11.8%

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

Alternative 10: 11.3% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * (s * u)) - 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. Add Preprocessing

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.2%

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

   1. mul-1-neg 25.2%

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

   2. unsub-neg 25.2%

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

8. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{-4 \cdot \frac{\left(u \cdot s\right) \cdot \left(\pi \cdot -0.5\right)}{s}}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Taylor expanded in s around 0 25.2%

   \[\leadsto \color{blue}{2 \cdot \left(s \cdot u\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
10. Step-by-step derivation

    1. *-commutative 25.2%

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

11. Simplified 25.2%

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

12. Taylor expanded in s around inf 11.6%

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

13. Final simplification 11.6%

    \[\leadsto 2 \cdot \left(s \cdot u\right) - \pi \]
14. Add Preprocessing

Alternative 11: 11.2% accurate, 216.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. Add Preprocessing

5. Taylor expanded in u around 0 11.6%

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

   1. neg-mul-1 11.6%

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

7. Simplified 11.6%

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

8. Final simplification 11.6%

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

Reproduce

herbie shell --seed 2024059 
(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))))