Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 14.2s

Alternatives: 7

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 97.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 85.2%

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

   1. +-commutative 85.2%

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

6. Simplified 85.2%

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

7. Taylor expanded in s around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. *-commutative 96.7%

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

   3. distribute-rgt-neg-in 96.7%

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

   4. associate--l+ 96.7%

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

   5. mul-1-neg 96.7%

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

   6. distribute-frac-neg 96.7%

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

   7. sub-neg 96.7%

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

   8. metadata-eval 96.7%

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

9. Simplified 96.7%

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

    1. flip-+ 96.7%

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

    2. metadata-eval 96.7%

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

11. Applied egg-rr 96.7%

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

    1. sub-neg 96.7%

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

    2. metadata-eval 96.7%

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

    3. +-commutative 96.7%

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

    4. associate-*r/ 96.7%

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

    5. *-rgt-identity 96.7%

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

    6. sub-neg 96.7%

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

    7. metadata-eval 96.7%

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

    8. +-commutative 96.7%

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

    9. *-lft-identity 96.7%

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

    10. metadata-eval 96.7%

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

    11. cancel-sign-sub-inv 96.7%

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

    12. *-commutative 96.7%

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

    13. associate-*l/ 96.7%

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

    14. metadata-eval 96.7%

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

13. Simplified 96.7%

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

14. Final simplification 96.7%

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

Alternative 3: 97.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 85.2%

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

   1. +-commutative 85.2%

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

6. Simplified 85.2%

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

7. Taylor expanded in s around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. *-commutative 96.7%

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

   3. distribute-rgt-neg-in 96.7%

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

   4. associate--l+ 96.7%

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

   5. mul-1-neg 96.7%

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

   6. distribute-frac-neg 96.7%

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

   7. sub-neg 96.7%

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

   8. metadata-eval 96.7%

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

9. Simplified 96.7%

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

10. Final simplification 96.7%

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

Alternative 4: 76.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 85.2%

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

   1. +-commutative 85.2%

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

6. Simplified 85.2%

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

7. Taylor expanded in s around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. *-commutative 96.7%

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

   3. distribute-rgt-neg-in 96.7%

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

   4. associate--l+ 96.7%

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

   5. mul-1-neg 96.7%

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

   6. distribute-frac-neg 96.7%

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

   7. sub-neg 96.7%

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

   8. metadata-eval 96.7%

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

9. Simplified 96.7%

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

10. Taylor expanded in u around 0 73.5%

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

11. Final simplification 73.5%

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

Alternative 5: 37.1% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -s * log(((-1.0e0) + (2.0e0 / u)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 85.2%

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

   1. +-commutative 85.2%

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

6. Simplified 85.2%

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

7. Taylor expanded in s around 0 96.7%

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

   1. mul-1-neg 96.7%

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

   2. *-commutative 96.7%

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

   3. distribute-rgt-neg-in 96.7%

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

   4. associate--l+ 96.7%

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

   5. mul-1-neg 96.7%

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

   6. distribute-frac-neg 96.7%

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

   7. sub-neg 96.7%

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

   8. metadata-eval 96.7%

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

9. Simplified 96.7%

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

10. Taylor expanded in s around inf 36.2%

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

    1. associate-*r* 36.2%

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

    2. neg-mul-1 36.2%

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

    3. sub-neg 36.2%

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

    4. associate-*r/ 36.2%

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

    5. metadata-eval 36.2%

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

    6. metadata-eval 36.2%

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

12. Simplified 36.2%

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

13. Final simplification 36.2%

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

Alternative 6: 16.1% accurate, 146.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = (-2.0e0) * (s * u)
end function

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(-2.0) * (s * u);
end

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 9.8%

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

5. Taylor expanded in u around 0 9.6%

   \[\leadsto s \cdot \left(-\color{blue}{-4 \cdot \left(u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - 0.5\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 9.6%

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

   2. associate-*l* 9.6%

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

   3. sub-neg 9.6%

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

   4. mul-1-neg 9.6%

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

   5. distribute-frac-neg 9.6%

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

   6. metadata-eval 9.6%

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

7. Simplified 9.6%

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

8. Taylor expanded in s around inf 16.0%

   \[\leadsto s \cdot \left(-u \cdot \left(\left(\frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{\pi}{s}\right)}} + -0.5\right) \cdot -4\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg 16.0%

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

   2. unsub-neg 16.0%

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

10. Simplified 16.0%

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

11. Taylor expanded in s around 0 16.0%

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

12. Final simplification 16.0%

    \[\leadsto -2 \cdot \left(s \cdot u\right) \]

Alternative 7: 16.1% accurate, 146.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * (u * (-2.0e0))
end function

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * (u * single(-2.0));
end

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 9.8%

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

5. Taylor expanded in u around 0 9.6%

   \[\leadsto s \cdot \left(-\color{blue}{-4 \cdot \left(u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - 0.5\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 9.6%

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

   2. associate-*l* 9.6%

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

   3. sub-neg 9.6%

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

   4. mul-1-neg 9.6%

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

   5. distribute-frac-neg 9.6%

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

   6. metadata-eval 9.6%

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

7. Simplified 9.6%

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

8. Taylor expanded in s around inf 16.0%

   \[\leadsto s \cdot \left(-u \cdot \left(\left(\frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{\pi}{s}\right)}} + -0.5\right) \cdot -4\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg 16.0%

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

   2. unsub-neg 16.0%

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

10. Simplified 16.0%

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

11. Taylor expanded in s around 0 16.0%

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

    1. *-commutative 16.0%

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

    2. associate-*l* 16.0%

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

13. Simplified 16.0%

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

14. Final simplification 16.0%

    \[\leadsto s \cdot \left(u \cdot -2\right) \]

Reproduce

herbie shell --seed 2023178 
(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))))