Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.9%

Time: 19.0s

Alternatives: 15

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * (log(sqrt((single(-1.0) + (single(1.0) / (((single(1.0) - u) / (single(1.0) + exp((single(pi) / s)))) + (u / (single(1.0) + exp(-(single(pi) / s))))))))) * -single(2.0));
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. clear-num 98.9%

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

   2. inv-pow 98.9%

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

6. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

8. Simplified 98.9%

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

   1. clear-num 98.9%

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

   2. add-sqr-sqrt 99.0%

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

   3. log-prod 99.0%

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

10. Applied egg-rr 99.0%

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

    1. count-2 99.0%

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

12. Simplified 99.0%

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

13. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

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(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(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 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. clear-num 98.9%

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

   2. inv-pow 98.9%

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

6. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 3: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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%

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

Alternative 4: 86.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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 84.8%

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

6. Final simplification 84.8%

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

Alternative 5: 25.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * ((log(s) - (u * single(-2.0))) - log(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 s around inf 25.0%

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

   1. +-commutative 25.0%

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

   2. fma-define 25.0%

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

7. Simplified 25.0%

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

8. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-/l* 25.1%

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

   3. associate-/r* 25.1%

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

10. Simplified 25.1%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right)} \]
11. Step-by-step derivation

    1. neg-sub0 25.1%

       \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    2. flip-- 21.5%

       \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    3. metadata-eval 21.5%

       \[\leadsto \frac{\color{blue}{0} - s \cdot s}{0 + s} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    4. pow2 21.5%

       \[\leadsto \frac{0 - \color{blue}{{s}^{2}}}{0 + s} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    5. add-sqr-sqrt 21.5%

       \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    6. sqrt-unprod 16.4%

       \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s \cdot s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    7. sqr-neg 16.4%

       \[\leadsto \frac{0 - {s}^{2}}{0 + \sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    8. sqrt-unprod -0.0%

       \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    9. add-sqr-sqrt 8.2%

       \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\left(-s\right)}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    10. sub-neg 8.2%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{0 - s}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    11. neg-sub0 8.2%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{-s}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    12. add-sqr-sqrt -0.0%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    13. sqrt-unprod 16.4%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    14. sqr-neg 16.4%

        \[\leadsto \frac{0 - {s}^{2}}{\sqrt{\color{blue}{s \cdot s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    15. sqrt-unprod 21.5%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    16. add-sqr-sqrt 21.5%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{s}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

12. Applied egg-rr 21.5%

    \[\leadsto \color{blue}{\frac{0 - {s}^{2}}{s}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]
13. Step-by-step derivation

    1. sub0-neg 21.5%

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

14. Simplified 21.5%

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

15. Taylor expanded in s around 0 25.4%

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

    1. mul-1-neg 25.4%

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

    2. *-commutative 25.4%

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

    3. distribute-rgt-neg-in 25.4%

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

    4. mul-1-neg 25.4%

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

    5. unsub-neg 25.4%

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

    6. *-commutative 25.4%

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

17. Simplified 25.4%

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

18. Final simplification 25.4%

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

Alternative 6: 25.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * (log(s) - log(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 s around inf 25.0%

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

   1. +-commutative 25.0%

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

   2. fma-define 25.0%

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

7. Simplified 25.0%

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

8. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. neg-mul-1 25.1%

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

   3. log1p-define 25.1%

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

10. Simplified 25.1%

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

11. Taylor expanded in s around 0 25.4%

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

    1. mul-1-neg 25.4%

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

    2. *-commutative 25.4%

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

    3. distribute-rgt-neg-in 25.4%

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

    4. mul-1-neg 25.4%

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

    5. unsub-neg 25.4%

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

13. Simplified 25.4%

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

14. Final simplification 25.4%

    \[\leadsto s \cdot \left(\log s - \log \pi\right) \]
15. Add Preprocessing

Alternative 7: 25.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(Float32(Float32(-2.0) * Float32(u * Float32(Float32(-1.0) + Float32(s / Float32(pi))))) - 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 25.0%

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

   1. +-commutative 25.0%

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

   2. fma-define 25.0%

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

7. Simplified 25.0%

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

8. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-/l* 25.1%

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

   3. associate-/r* 25.1%

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

10. Simplified 25.1%

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

11. Taylor expanded in s around 0 25.1%

    \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \color{blue}{\left(1 + -1 \cdot \frac{s}{\pi}\right)}\right)\right) \]
12. Step-by-step derivation

    1. mul-1-neg 25.1%

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

    2. distribute-frac-neg2 25.1%

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

13. Simplified 25.1%

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

14. Final simplification 25.1%

    \[\leadsto s \cdot \left(-2 \cdot \left(u \cdot \left(-1 + \frac{s}{\pi}\right)\right) - \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]
15. Add Preprocessing

Alternative 8: 25.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) * Float32(Float32(1.0) / 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 25.0%

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

   1. +-commutative 25.0%

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

   2. fma-define 25.0%

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

7. Simplified 25.0%

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

8. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. neg-mul-1 25.1%

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

   3. log1p-define 25.1%

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

10. Simplified 25.1%

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

    1. clear-num 25.1%

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

    2. associate-/r/ 25.1%

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

12. Applied egg-rr 25.1%

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

13. Final simplification 25.1%

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

Alternative 9: 25.1% accurate, 4.1× speedup

Math

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

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(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 25.0%

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

   1. +-commutative 25.0%

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

   2. fma-define 25.0%

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

7. Simplified 25.0%

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

8. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. neg-mul-1 25.1%

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

   3. log1p-define 25.1%

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

10. Simplified 25.1%

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

Alternative 10: 11.6% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(0.25) * (u * single(pi))) - (u * ((single(pi) * single(-0.25)) + (single(0.25) * (single(pi) / u)))));
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 11.8%

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

6. Taylor expanded in u around inf 11.8%

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

7. Final simplification 11.8%

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

Alternative 11: 11.6% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = u * ((single(4.0) * (single(pi) * single(0.5))) - (single(pi) / u));
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 11.8%

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

6. Taylor expanded in u around inf 11.8%

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

   1. +-commutative 11.8%

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

   2. mul-1-neg 11.8%

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

   3. unsub-neg 11.8%

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

   4. distribute-rgt-out-- 11.8%

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

   5. metadata-eval 11.8%

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

8. Simplified 11.8%

   \[\leadsto \color{blue}{u \cdot \left(4 \cdot \left(\pi \cdot 0.5\right) - \frac{\pi}{u}\right)} \]
9. Add Preprocessing

Alternative 12: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(pi) * (single(2.0) * u)) - 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 s around inf 25.0%

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

   1. +-commutative 25.0%

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

   2. fma-define 25.0%

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

7. Simplified 25.0%

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

8. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-/l* 25.1%

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

   3. associate-/r* 25.1%

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

10. Simplified 25.1%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right)} \]
11. Step-by-step derivation

    1. neg-sub0 25.1%

       \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    2. flip-- 21.5%

       \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    3. metadata-eval 21.5%

       \[\leadsto \frac{\color{blue}{0} - s \cdot s}{0 + s} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    4. pow2 21.5%

       \[\leadsto \frac{0 - \color{blue}{{s}^{2}}}{0 + s} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    5. add-sqr-sqrt 21.5%

       \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    6. sqrt-unprod 16.4%

       \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s \cdot s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    7. sqr-neg 16.4%

       \[\leadsto \frac{0 - {s}^{2}}{0 + \sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    8. sqrt-unprod -0.0%

       \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    9. add-sqr-sqrt 8.2%

       \[\leadsto \frac{0 - {s}^{2}}{0 + \color{blue}{\left(-s\right)}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    10. sub-neg 8.2%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{0 - s}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    11. neg-sub0 8.2%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{-s}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    12. add-sqr-sqrt -0.0%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    13. sqrt-unprod 16.4%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    14. sqr-neg 16.4%

        \[\leadsto \frac{0 - {s}^{2}}{\sqrt{\color{blue}{s \cdot s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    15. sqrt-unprod 21.5%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

    16. add-sqr-sqrt 21.5%

        \[\leadsto \frac{0 - {s}^{2}}{\color{blue}{s}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]

12. Applied egg-rr 21.5%

    \[\leadsto \color{blue}{\frac{0 - {s}^{2}}{s}} \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + -2 \cdot \left(u \cdot \frac{\frac{\pi}{s}}{1 + \frac{\pi}{s}}\right)\right) \]
13. Step-by-step derivation

    1. sub0-neg 21.5%

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

14. Simplified 21.5%

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

15. Taylor expanded in s around -inf 11.8%

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

    1. +-commutative 11.8%

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

    2. mul-1-neg 11.8%

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

    3. unsub-neg 11.8%

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

    4. associate-*r* 11.8%

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

17. Simplified 11.8%

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

18. Final simplification 11.8%

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

Alternative 13: 11.4% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (s * -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 u around 0 11.6%

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

   1. associate-*r/ 11.6%

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

7. Applied egg-rr 11.6%

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

8. Final simplification 11.6%

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

Alternative 14: 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 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. Add Preprocessing

Alternative 15: 4.6% accurate, 433.0× 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(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 11.6%

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

   1. neg-sub0 11.6%

      \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \frac{\pi}{s} \]

   2. sub-neg 11.6%

      \[\leadsto \color{blue}{\left(0 + \left(-s\right)\right)} \cdot \frac{\pi}{s} \]

   3. add-sqr-sqrt -0.0%

      \[\leadsto \left(0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}\right) \cdot \frac{\pi}{s} \]

   4. sqrt-unprod 7.5%

      \[\leadsto \left(0 + \color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}\right) \cdot \frac{\pi}{s} \]

   5. sqr-neg 7.5%

      \[\leadsto \left(0 + \sqrt{\color{blue}{s \cdot s}}\right) \cdot \frac{\pi}{s} \]

   6. sqrt-unprod 4.5%

      \[\leadsto \left(0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}\right) \cdot \frac{\pi}{s} \]

   7. add-sqr-sqrt 4.5%

      \[\leadsto \left(0 + \color{blue}{s}\right) \cdot \frac{\pi}{s} \]

7. Applied egg-rr 4.5%

   \[\leadsto \color{blue}{\left(0 + s\right)} \cdot \frac{\pi}{s} \]
8. Step-by-step derivation

   1. +-lft-identity 4.5%

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

9. Simplified 4.5%

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

10. Taylor expanded in s around 0 4.5%

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

Reproduce

herbie shell --seed 2024145 
(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))))