Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.3%

Time: 13.0s

Alternatives: 9

Speedup: 1.3×

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: 99.0% 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: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{s} + 2\\ s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\frac{\pi}{s} \cdot t\_0}{t\_0}}}} + -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ (/ PI s) 2.0)))
   (*
    s
    (-
     (log
      (+
       (/
        1.0
        (+
         (/ u (+ 1.0 (exp (- (/ PI s)))))
         (/ (- 1.0 u) (+ 1.0 (exp (/ (* (/ PI s) t_0) t_0))))))
       -1.0))))))

C

float code(float u, float s) {
	float t_0 = (((float) M_PI) / s) + 2.0f;
	return s * -logf(((1.0f / ((u / (1.0f + expf(-(((float) M_PI) / s)))) + ((1.0f - u) / (1.0f + expf((((((float) M_PI) / s) * t_0) / t_0)))))) + -1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(Float32(pi) / s) + Float32(2.0))
	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(Float32(Float32(pi) / s) * t_0) / t_0)))))) + Float32(-1.0)))))
end

MATLAB

function tmp = code(u, s)
	t_0 = (single(pi) / s) + single(2.0);
	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) * t_0) / t_0)))))) + single(-1.0)));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

   1. expm1-log1p-u 98.9%

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

   2. expm1-undefine 98.9%

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

6. Applied egg-rr 98.9%

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

   1. expm1-define 98.9%

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

8. Simplified 98.9%

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

   1. expm1-undefine 98.9%

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

   2. flip-- 98.9%

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

   3. log1p-undefine 98.9%

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

   4. rem-exp-log 98.9%

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

   5. +-commutative 98.9%

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

   6. log1p-undefine 98.9%

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

   7. rem-exp-log 98.9%

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

   8. +-commutative 98.9%

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

   9. metadata-eval 98.9%

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

   10. log1p-undefine 98.9%

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

   11. rem-exp-log 98.9%

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

   12. +-commutative 98.9%

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

10. Applied egg-rr 98.9%

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

    1. difference-of-sqr-1 98.9%

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

    2. associate-+l+ 98.9%

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

    3. metadata-eval 98.9%

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

    4. associate--l+ 98.9%

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

    5. metadata-eval 98.9%

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\left(\frac{\pi}{s} + 2\right) \cdot \left(\frac{\pi}{s} + \color{blue}{0}\right)}{\left(\frac{\pi}{s} + 1\right) + 1}}}} + -1\right) \]

    6. associate-+l+ 98.9%

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\left(\frac{\pi}{s} + 2\right) \cdot \left(\frac{\pi}{s} + 0\right)}{\color{blue}{\frac{\pi}{s} + \left(1 + 1\right)}}}}} + -1\right) \]

    7. metadata-eval 98.9%

       \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\left(\frac{\pi}{s} + 2\right) \cdot \left(\frac{\pi}{s} + 0\right)}{\frac{\pi}{s} + \color{blue}{2}}}}} + -1\right) \]

12. Simplified 98.9%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\frac{\left(\frac{\pi}{s} + 2\right) \cdot \left(\frac{\pi}{s} + 0\right)}{\frac{\pi}{s} + 2}}}}} + -1\right) \]

13. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

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(Float32(-s) * 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 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 25.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.25 \cdot \left(u \cdot \pi\right) - \pi \cdot -0.25\\ s \cdot \left(\left(\log s - \log \left(4 \cdot t\_0\right)\right) + -0.25 \cdot \frac{s}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (- (* -0.25 (* u PI)) (* PI -0.25))))
   (* s (+ (- (log s) (log (* 4.0 t_0))) (* -0.25 (/ s t_0))))))

C

float code(float u, float s) {
	float t_0 = (-0.25f * (u * ((float) M_PI))) - (((float) M_PI) * -0.25f);
	return s * ((logf(s) - logf((4.0f * t_0))) + (-0.25f * (s / t_0)));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(Float32(-0.25) * Float32(u * Float32(pi))) - Float32(Float32(pi) * Float32(-0.25)))
	return Float32(s * Float32(Float32(log(s) - log(Float32(Float32(4.0) * t_0))) + Float32(Float32(-0.25) * Float32(s / t_0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = (single(-0.25) * (u * single(pi))) - (single(pi) * single(-0.25));
	tmp = s * ((log(s) - log((single(4.0) * t_0))) + (single(-0.25) * (s / t_0)));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 25.3%

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

   1. *-commutative 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in s around 0 25.3%

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

10. Final simplification 25.3%

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

Alternative 4: 25.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\pi \cdot \left(u \cdot -0.25 - -0.25\right)}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (+ 1.0 (* 4.0 (/ (* PI (- (* u -0.25) -0.25)) s))))))

C

float code(float u, float s) {
	return -s * logf((1.0f + (4.0f * ((((float) M_PI) * ((u * -0.25f) - -0.25f)) / s))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(1.0) + Float32(Float32(4.0) * Float32(Float32(Float32(pi) * Float32(Float32(u * Float32(-0.25)) - Float32(-0.25))) / s)))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(1.0) + (single(4.0) * ((single(pi) * ((u * single(-0.25)) - single(-0.25))) / s))));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 25.3%

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

   1. *-commutative 25.3%

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

8. Simplified 25.3%

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

   1. associate-*r* 25.3%

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

   2. *-commutative 25.3%

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

   3. distribute-rgt-out-- 25.3%

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

10. Applied egg-rr 25.3%

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

11. Final simplification 25.3%

    \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{\pi \cdot \left(u \cdot -0.25 - -0.25\right)}{s}\right) \]
12. Add Preprocessing

Alternative 5: 25.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 25.3%

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

   1. *-commutative 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in u around 0 25.3%

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

    1. +-commutative 25.3%

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

    2. mul-1-neg 25.3%

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

    3. unsub-neg 25.3%

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

    4. associate-/l* 25.3%

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

    5. +-commutative 25.3%

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

    6. log1p-define 25.3%

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

11. Simplified 25.3%

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

12. Final simplification 25.3%

    \[\leadsto u \cdot \frac{\pi}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
13. Add Preprocessing

Alternative 6: 25.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 25.3%

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

   1. *-commutative 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in u around 0 25.3%

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

10. Final simplification 25.3%

    \[\leadsto \left(-s\right) \cdot \log \left(1 + \frac{\pi}{s}\right) \]
11. 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 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 25.3%

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

   1. *-commutative 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in u around 0 25.3%

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

    1. mul-1-neg 25.3%

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

    2. *-commutative 25.3%

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

    3. distribute-rgt-neg-in 25.3%

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

    4. log1p-define 25.3%

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

11. Simplified 25.3%

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

12. Final simplification 25.3%

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

Alternative 8: 12.3% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 11.0%

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

   1. associate--r+ 11.0%

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

   2. cancel-sign-sub-inv 11.0%

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

   3. metadata-eval 11.0%

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

   4. cancel-sign-sub-inv 11.0%

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

   5. associate-*r* 11.0%

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

   6. distribute-rgt-out 11.0%

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

   7. *-commutative 11.0%

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

   8. metadata-eval 11.0%

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

   9. *-commutative 11.0%

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

   10. associate-*l* 11.0%

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

7. Simplified 11.0%

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

8. Taylor expanded in u around inf 5.0%

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

   1. associate-*r* 5.0%

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

10. Simplified 5.0%

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

    1. add-sqr-sqrt -0.0%

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

    2. sqrt-unprod 12.1%

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

    3. pow2 12.1%

       \[\leadsto -4 \cdot \sqrt{\color{blue}{{\left(\left(-0.5 \cdot u\right) \cdot \pi\right)}^{2}}} \]

    4. *-commutative 12.1%

       \[\leadsto -4 \cdot \sqrt{{\color{blue}{\left(\pi \cdot \left(-0.5 \cdot u\right)\right)}}^{2}} \]

12. Applied egg-rr 12.1%

    \[\leadsto -4 \cdot \color{blue}{\sqrt{{\left(\pi \cdot \left(-0.5 \cdot u\right)\right)}^{2}}} \]

13. Taylor expanded in u around 0 12.1%

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

    1. *-commutative 12.1%

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

    2. *-commutative 12.1%

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

15. Simplified 12.1%

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

16. Final simplification 12.1%

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

Alternative 9: 11.4% 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 10.8%

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

   1. neg-mul-1 10.8%

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

7. Simplified 10.8%

   \[\leadsto \color{blue}{-\pi} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024092 
(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))))