Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 11.5s

Alternatives: 10

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: 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: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.8%

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

   1. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 3: 25.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 24.1%

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

5. Taylor expanded in u around 0 24.6%

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

   1. *-commutative 24.6%

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

   2. associate-/r/ 24.6%

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

7. Simplified 24.6%

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

8. Taylor expanded in s around 0 24.6%

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

9. Final simplification 24.6%

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

Alternative 4: 25.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 24.1%

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

5. Taylor expanded in u around 0 24.6%

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

   1. *-commutative 24.6%

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

   2. associate-/r/ 24.6%

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

7. Simplified 24.6%

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

8. Taylor expanded in s around 0 24.6%

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

9. Taylor expanded in s around 0 24.6%

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

    1. +-commutative 24.6%

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

    2. mul-1-neg 24.6%

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

    3. unsub-neg 24.6%

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

    4. *-commutative 24.6%

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

    5. neg-mul-1 24.6%

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

    6. +-commutative 24.6%

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

    7. log-rec 24.6%

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

    8. log-rec 24.6%

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

    9. sub-neg 24.6%

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

    10. log-div 24.6%

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

    11. associate-*r* 24.6%

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

    12. neg-mul-1 24.6%

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

    13. unpow2 24.6%

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

    14. associate-*l/ 24.6%

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

11. Simplified 24.6%

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

12. Final simplification 24.6%

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

Alternative 5: 25.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \mathsf{log1p}\left(\frac{-0.25}{s} \cdot \left(\pi \cdot -4\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log1p (* (/ -0.25 s) (* PI -4.0)))))

C

float code(float u, float s) {
	return -s * log1pf(((-0.25f / s) * (((float) M_PI) * -4.0f)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(Float32(-0.25) / s) * Float32(Float32(pi) * Float32(-4.0)))))
end

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 24.1%

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

5. Taylor expanded in u around 0 24.6%

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

   1. *-commutative 24.6%

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

   2. associate-/r/ 24.6%

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

7. Simplified 24.6%

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

   1. distribute-rgt-neg-out 24.6%

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

   2. neg-sub0 24.6%

      \[\leadsto \color{blue}{0 - s \cdot \log \left(-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1\right)} \]

   3. +-commutative 24.6%

      \[\leadsto 0 - s \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{\pi}{\frac{s}{-0.25}}\right)} \]

   4. log1p-udef 24.6%

      \[\leadsto 0 - s \cdot \color{blue}{\mathsf{log1p}\left(-4 \cdot \frac{\pi}{\frac{s}{-0.25}}\right)} \]

   5. associate-*r/ 24.6%

      \[\leadsto 0 - s \cdot \mathsf{log1p}\left(\color{blue}{\frac{-4 \cdot \pi}{\frac{s}{-0.25}}}\right) \]

   6. div-inv 24.6%

      \[\leadsto 0 - s \cdot \mathsf{log1p}\left(\color{blue}{\left(-4 \cdot \pi\right) \cdot \frac{1}{\frac{s}{-0.25}}}\right) \]

   7. clear-num 24.6%

      \[\leadsto 0 - s \cdot \mathsf{log1p}\left(\left(-4 \cdot \pi\right) \cdot \color{blue}{\frac{-0.25}{s}}\right) \]

9. Applied egg-rr 24.6%

   \[\leadsto \color{blue}{0 - s \cdot \mathsf{log1p}\left(\left(-4 \cdot \pi\right) \cdot \frac{-0.25}{s}\right)} \]
10. Step-by-step derivation

    1. neg-sub0 24.6%

       \[\leadsto \color{blue}{-s \cdot \mathsf{log1p}\left(\left(-4 \cdot \pi\right) \cdot \frac{-0.25}{s}\right)} \]

    2. distribute-lft-neg-in 24.6%

       \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(\left(-4 \cdot \pi\right) \cdot \frac{-0.25}{s}\right)} \]

    3. *-commutative 24.6%

       \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\color{blue}{\frac{-0.25}{s} \cdot \left(-4 \cdot \pi\right)}\right) \]

    4. *-commutative 24.6%

       \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\frac{-0.25}{s} \cdot \color{blue}{\left(\pi \cdot -4\right)}\right) \]

11. Simplified 24.6%

    \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(\frac{-0.25}{s} \cdot \left(\pi \cdot -4\right)\right)} \]

12. Final simplification 24.6%

    \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\frac{-0.25}{s} \cdot \left(\pi \cdot -4\right)\right) \]

Alternative 6: 25.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ -3 \cdot \left(s \cdot \left(0.3333333333333333 \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* -3.0 (* s (* 0.3333333333333333 (log1p (/ PI s))))))

C

float code(float u, float s) {
	return -3.0f * (s * (0.3333333333333333f * log1pf((((float) M_PI) / s))));
}

Julia

function code(u, s)
	return Float32(Float32(-3.0) * Float32(s * Float32(Float32(0.3333333333333333) * log1p(Float32(Float32(pi) / s)))))
end

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 24.1%

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

5. Taylor expanded in u around 0 24.6%

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

   1. *-commutative 24.6%

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

   2. associate-/r/ 24.6%

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

7. Simplified 24.6%

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

   1. add-cube-cbrt 24.6%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(\left(\sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1} \cdot \sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1}\right) \cdot \sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1}\right)}\right) \]

   2. log-prod 24.6%

      \[\leadsto s \cdot \left(-\color{blue}{\left(\log \left(\sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1} \cdot \sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1}\right) + \log \left(\sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1}\right)\right)}\right) \]

   3. pow2 24.6%

      \[\leadsto s \cdot \left(-\left(\log \color{blue}{\left({\left(\sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1}\right)}^{2}\right)} + \log \left(\sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1}\right)\right)\right) \]

   4. *-commutative 24.6%

      \[\leadsto s \cdot \left(-\left(\log \left({\left(\sqrt[3]{\color{blue}{\frac{\pi}{\frac{s}{-0.25}} \cdot -4} + 1}\right)}^{2}\right) + \log \left(\sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1}\right)\right)\right) \]

   5. fma-def 24.6%

      \[\leadsto s \cdot \left(-\left(\log \left({\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(\frac{\pi}{\frac{s}{-0.25}}, -4, 1\right)}}\right)}^{2}\right) + \log \left(\sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1}\right)\right)\right) \]

   6. div-inv 24.6%

      \[\leadsto s \cdot \left(-\left(\log \left({\left(\sqrt[3]{\mathsf{fma}\left(\color{blue}{\pi \cdot \frac{1}{\frac{s}{-0.25}}}, -4, 1\right)}\right)}^{2}\right) + \log \left(\sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1}\right)\right)\right) \]

   7. clear-num 24.6%

      \[\leadsto s \cdot \left(-\left(\log \left({\left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot \color{blue}{\frac{-0.25}{s}}, -4, 1\right)}\right)}^{2}\right) + \log \left(\sqrt[3]{-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1}\right)\right)\right) \]

9. Applied egg-rr 24.6%

   \[\leadsto s \cdot \left(-\color{blue}{\left(\log \left({\left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot \frac{-0.25}{s}, -4, 1\right)}\right)}^{2}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot \frac{-0.25}{s}, -4, 1\right)}\right)\right)}\right) \]
10. Step-by-step derivation

    1. log-pow 24.6%

       \[\leadsto s \cdot \left(-\left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot \frac{-0.25}{s}, -4, 1\right)}\right)} + \log \left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot \frac{-0.25}{s}, -4, 1\right)}\right)\right)\right) \]

    2. distribute-lft1-in 24.6%

       \[\leadsto s \cdot \left(-\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot \frac{-0.25}{s}, -4, 1\right)}\right)}\right) \]

    3. metadata-eval 24.6%

       \[\leadsto s \cdot \left(-\color{blue}{3} \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\pi \cdot \frac{-0.25}{s}, -4, 1\right)}\right)\right) \]

    4. associate-*r/ 24.6%

       \[\leadsto s \cdot \left(-3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\pi \cdot -0.25}{s}}, -4, 1\right)}\right)\right) \]

    5. associate-*l/ 24.6%

       \[\leadsto s \cdot \left(-3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{\pi}{s} \cdot -0.25}, -4, 1\right)}\right)\right) \]

    6. *-commutative 24.6%

       \[\leadsto s \cdot \left(-3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\color{blue}{-0.25 \cdot \frac{\pi}{s}}, -4, 1\right)}\right)\right) \]

11. Simplified 24.6%

    \[\leadsto s \cdot \left(-\color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(-0.25 \cdot \frac{\pi}{s}, -4, 1\right)}\right)}\right) \]
12. Step-by-step derivation

    1. expm1-log1p-u 24.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(-3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(-0.25 \cdot \frac{\pi}{s}, -4, 1\right)}\right)\right)\right)\right)} \]

    2. expm1-udef 15.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(-3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(-0.25 \cdot \frac{\pi}{s}, -4, 1\right)}\right)\right)\right)} - 1} \]

    3. distribute-lft-neg-in 15.4%

       \[\leadsto e^{\mathsf{log1p}\left(s \cdot \color{blue}{\left(\left(-3\right) \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(-0.25 \cdot \frac{\pi}{s}, -4, 1\right)}\right)\right)}\right)} - 1 \]

    4. metadata-eval 15.4%

       \[\leadsto e^{\mathsf{log1p}\left(s \cdot \left(\color{blue}{-3} \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(-0.25 \cdot \frac{\pi}{s}, -4, 1\right)}\right)\right)\right)} - 1 \]

    5. associate-*r/ 15.4%

       \[\leadsto e^{\mathsf{log1p}\left(s \cdot \left(-3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\color{blue}{\frac{-0.25 \cdot \pi}{s}}, -4, 1\right)}\right)\right)\right)} - 1 \]

13. Applied egg-rr 15.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(s \cdot \left(-3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\frac{-0.25 \cdot \pi}{s}, -4, 1\right)}\right)\right)\right)} - 1} \]
14. Step-by-step derivation

    1. expm1-def 24.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(-3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\frac{-0.25 \cdot \pi}{s}, -4, 1\right)}\right)\right)\right)\right)} \]

    2. expm1-log1p 24.6%

       \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\frac{-0.25 \cdot \pi}{s}, -4, 1\right)}\right)\right)} \]

    3. *-commutative 24.6%

       \[\leadsto \color{blue}{\left(-3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(\frac{-0.25 \cdot \pi}{s}, -4, 1\right)}\right)\right) \cdot s} \]

    4. associate-*l* 24.6%

       \[\leadsto \color{blue}{-3 \cdot \left(\log \left(\sqrt[3]{\mathsf{fma}\left(\frac{-0.25 \cdot \pi}{s}, -4, 1\right)}\right) \cdot s\right)} \]

15. Simplified 24.6%

    \[\leadsto \color{blue}{-3 \cdot \left(\left(0.3333333333333333 \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \cdot s\right)} \]

16. Final simplification 24.6%

    \[\leadsto -3 \cdot \left(s \cdot \left(0.3333333333333333 \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)\right)\right) \]

Alternative 7: 25.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 24.1%

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

5. Taylor expanded in u around 0 24.6%

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

   1. *-commutative 24.6%

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

   2. associate-/r/ 24.6%

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

7. Simplified 24.6%

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

8. Taylor expanded in s around 0 24.6%

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

   1. mul-1-neg 24.6%

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

   2. *-commutative 24.6%

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

   3. distribute-rgt-neg-in 24.6%

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

   4. +-commutative 24.6%

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

   5. mul-1-neg 24.6%

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

   6. sub-neg 24.6%

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

   7. log-div 24.6%

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

10. Simplified 24.6%

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

11. Final simplification 24.6%

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

Alternative 8: 12.8% accurate, 7.0× 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(Float32(-s) / Float32(pi)))
end

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 24.1%

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

5. Taylor expanded in u around 0 24.6%

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

   1. *-commutative 24.6%

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

   2. associate-/r/ 24.6%

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

7. Simplified 24.6%

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

8. Taylor expanded in s around 0 24.6%

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

9. Taylor expanded in s around inf 12.7%

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

10. Final simplification 12.7%

    \[\leadsto s \cdot \frac{-s}{\pi} \]

Alternative 9: 11.4% accurate, 7.2× 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 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in u around 0 11.5%

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

   1. mul-1-neg 11.5%

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

6. Simplified 11.5%

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

7. Final simplification 11.5%

   \[\leadsto -\pi \]

Alternative 10: 4.6% accurate, 7.3× 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 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 24.1%

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

5. Taylor expanded in u around 0 24.6%

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

   1. *-commutative 24.6%

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

   2. associate-/r/ 24.6%

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

7. Simplified 24.6%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 8.9%

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

   3. pow2 8.9%

      \[\leadsto \sqrt{\color{blue}{{\left(s \cdot \left(-\log \left(-4 \cdot \frac{\pi}{\frac{s}{-0.25}} + 1\right)\right)\right)}^{2}}} \]

9. Applied egg-rr 8.9%

   \[\leadsto \color{blue}{\sqrt{{\left(s \cdot \mathsf{log1p}\left(\left(-4 \cdot \pi\right) \cdot \frac{-0.25}{s}\right)\right)}^{2}}} \]
10. Step-by-step derivation

    1. *-commutative 8.9%

       \[\leadsto \sqrt{{\left(s \cdot \mathsf{log1p}\left(\color{blue}{\frac{-0.25}{s} \cdot \left(-4 \cdot \pi\right)}\right)\right)}^{2}} \]

    2. *-commutative 8.9%

       \[\leadsto \sqrt{{\left(s \cdot \mathsf{log1p}\left(\frac{-0.25}{s} \cdot \color{blue}{\left(\pi \cdot -4\right)}\right)\right)}^{2}} \]

11. Simplified 8.9%

    \[\leadsto \color{blue}{\sqrt{{\left(s \cdot \mathsf{log1p}\left(\frac{-0.25}{s} \cdot \left(\pi \cdot -4\right)\right)\right)}^{2}}} \]

12. Taylor expanded in s around inf 4.7%

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

13. Final simplification 4.7%

    \[\leadsto \pi \]

Reproduce

herbie shell --seed 2023222 
(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))))