Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.9%

Time: 15.6s

Alternatives: 9

Speedup: N/A×

Specification

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 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, 0.8× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{u + -1}{-1 - e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     -1.0
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (+ u -1.0) (- -1.0 (exp (pow (cbrt (/ PI s)) 3.0)))))))))))

C

float code(float u, float s) {
	return s * -logf((-1.0f + (1.0f / ((u / (1.0f + expf((((float) M_PI) / -s)))) + ((u + -1.0f) / (-1.0f - expf(powf(cbrtf((((float) M_PI) / s)), 3.0f))))))));
}

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(pi) / Float32(-s))))) + Float32(Float32(u + Float32(-1.0)) / Float32(Float32(-1.0) - exp((cbrt(Float32(Float32(pi) / s)) ^ Float32(3.0)))))))))))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

   1. add-cube-cbrt 99.1%

      \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{u + -1}{-1 - e^{\color{blue}{\left(\sqrt[3]{\frac{\pi}{s}} \cdot \sqrt[3]{\frac{\pi}{s}}\right) \cdot \sqrt[3]{\frac{\pi}{s}}}}}} + -1\right)\right) \]

   2. pow3 99.1%

      \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{u + -1}{-1 - e^{\color{blue}{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}} + -1\right)\right) \]

6. Applied egg-rr 99.1%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{u + -1}{-1 - e^{\color{blue}{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}} + -1\right)\right) \]

7. Final simplification 99.1%

   \[\leadsto s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{u + -1}{-1 - e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}\right)\right) \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     -1.0
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (+ u -1.0) (- -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)))) + ((u + -1.0f) / (-1.0f - expf((((float) M_PI) / s))))))));
}

Julia

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 3: 25.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 5.3%

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

6. Taylor expanded in s around inf 25.3%

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

   1. +-commutative 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in s around 0 25.5%

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

10. Taylor expanded in s around inf 25.4%

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

    1. log-rec 25.5%

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

    2. mul-1-neg 25.5%

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

    3. +-commutative 25.5%

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

    4. mul-1-neg 25.5%

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

    5. unsub-neg 25.5%

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

12. Simplified 25.5%

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

13. Final simplification 25.5%

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

Alternative 4: 25.3% 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 5.3%

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

6. Taylor expanded in s around inf 25.3%

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

   1. +-commutative 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in s around 0 25.5%

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

10. Taylor expanded in s around 0 25.5%

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

    1. mul-1-neg 25.5%

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

    2. sub-neg 25.5%

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

12. Simplified 25.5%

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

13. Final simplification 25.5%

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

Alternative 5: 25.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 5.3%

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

6. Taylor expanded in s around inf 25.3%

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

   1. +-commutative 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in s around 0 25.5%

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

    1. associate-*r* 25.5%

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

    2. neg-mul-1 25.5%

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

    3. mul-1-neg 25.5%

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

11. Simplified 25.5%

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

    1. distribute-lft-in 25.5%

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

    2. *-commutative 25.5%

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

    3. add-sqr-sqrt -0.0%

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

    4. sqrt-unprod 25.4%

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

    5. sqr-neg 25.4%

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

    6. sqrt-unprod 25.4%

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

    7. add-sqr-sqrt 25.4%

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

    8. add-sqr-sqrt -0.0%

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

    9. sqrt-unprod 7.5%

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

    10. sqr-neg 7.5%

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

    11. sqrt-unprod 7.2%

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

    12. add-sqr-sqrt 7.2%

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

    13. add-sqr-sqrt 7.2%

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

    14. sqrt-unprod 7.2%

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

    15. sqr-neg 7.2%

        \[\leadsto \log \pi \cdot s + s \cdot \sqrt{\color{blue}{\log s \cdot \log s}} \]

    16. sqrt-unprod -0.0%

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

    17. add-sqr-sqrt 25.4%

        \[\leadsto \log \pi \cdot s + s \cdot \color{blue}{\log s} \]

13. Applied egg-rr 25.4%

    \[\leadsto \color{blue}{\log \pi \cdot s + s \cdot \log s} \]
14. Step-by-step derivation

    1. *-commutative 25.4%

       \[\leadsto \color{blue}{s \cdot \log \pi} + s \cdot \log s \]

    2. distribute-lft-in 25.4%

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

    3. log-prod 25.4%

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

    4. *-commutative 25.4%

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

15. Simplified 25.4%

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

16. Final simplification 25.4%

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

Alternative 6: 25.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 5.3%

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

6. Taylor expanded in s around inf 25.3%

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

   1. +-commutative 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in s around 0 25.5%

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

    1. associate-*r* 25.5%

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

    2. neg-mul-1 25.5%

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

    3. mul-1-neg 25.5%

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

11. Simplified 25.5%

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

    1. add0 25.5%

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

    2. distribute-lft-neg-out 25.5%

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

    3. distribute-rgt-neg-in 25.5%

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

    4. unsub-neg 25.5%

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

    5. diff-log 25.3%

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

    6. log-rec 25.3%

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

    7. clear-num 25.5%

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

13. Applied egg-rr 25.5%

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

    1. add0 25.5%

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

15. Simplified 25.5%

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

16. Final simplification 25.5%

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

Alternative 7: 12.8% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 5.3%

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

6. Taylor expanded in s around inf 25.3%

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

   1. +-commutative 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in s around 0 25.5%

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

10. Taylor expanded in s around inf 12.7%

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

11. Final simplification 12.7%

    \[\leadsto s \cdot \frac{s}{-\pi} \]
12. Add Preprocessing

Alternative 8: 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 11.1%

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

   1. neg-mul-1 11.1%

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

7. Simplified 11.1%

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

8. Final simplification 11.1%

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

Alternative 9: 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 5.3%

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

   1. rem-log-exp 11.0%

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

   2. add-sqr-sqrt -0.0%

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

   3. sqrt-unprod 3.2%

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

   4. sqr-neg 3.2%

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

   5. sqrt-unprod 4.5%

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

   6. add-sqr-sqrt 4.5%

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

   7. associate-*r/ 4.5%

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

7. Applied egg-rr 4.5%

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

8. Taylor expanded in s around 0 4.5%

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

9. Final simplification 4.5%

   \[\leadsto \pi \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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))))