Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 99.0%

Time: 14.0s

Alternatives: 14

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 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: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 25.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in s around 0 25.0%

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

7. Taylor expanded in u around 0 25.2%

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

8. Final simplification 25.2%

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

Alternative 3: 25.1% 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. neg-mul-1 25.1%

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

   3. log1p-define 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in s around 0 25.2%

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

    1. neg-mul-1 25.2%

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

    2. +-commutative 25.2%

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

    3. unsub-neg 25.2%

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

11. Simplified 25.2%

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

12. Final simplification 25.2%

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

Alternative 4: 25.1% accurate, 2.1× speedup

Math

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

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(Float32(log(s) - 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%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. neg-mul-1 25.1%

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

   3. log1p-define 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in s around 0 25.2%

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

    1. mul-1-neg 25.2%

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

    2. unsub-neg 25.2%

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

    3. mul-1-neg 25.2%

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

    4. neg-mul-1 25.2%

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

    5. +-commutative 25.2%

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

    6. distribute-neg-in 25.2%

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

    7. remove-double-neg 25.2%

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

11. Simplified 25.2%

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

12. Final simplification 25.2%

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

Alternative 5: 25.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. neg-mul-1 25.1%

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

   3. log1p-define 25.1%

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

8. Simplified 25.1%

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

9. Taylor expanded in s around 0 25.2%

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

    1. neg-mul-1 25.2%

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

    2. unsub-neg 25.2%

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

11. Simplified 25.2%

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

12. Final simplification 25.2%

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

Alternative 6: 24.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \left(\left(s \cdot u\right) \cdot \frac{\pi \cdot -0.5}{s}\right)}{\frac{\pi}{s} + 1} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (-
  (/ (* -4.0 (* (* s u) (/ (* PI -0.5) s))) (+ (/ PI s) 1.0))
  (* s (log1p (/ PI s)))))

C

float code(float u, float s) {
	return ((-4.0f * ((s * u) * ((((float) M_PI) * -0.5f) / s))) / ((((float) M_PI) / s) + 1.0f)) - (s * log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(Float32(Float32(-4.0) * Float32(Float32(s * u) * Float32(Float32(Float32(pi) * Float32(-0.5)) / s))) / Float32(Float32(Float32(pi) / s) + Float32(1.0))) - Float32(s * log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. mul-1-neg 25.1%

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

   2. unsub-neg 25.1%

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

8. Simplified 25.1%

   \[\leadsto \color{blue}{\frac{-4 \cdot \left(\left(s \cdot u\right) \cdot \frac{\pi \cdot -0.5}{s}\right)}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

9. Final simplification 25.1%

   \[\leadsto \frac{-4 \cdot \left(\left(s \cdot u\right) \cdot \frac{\pi \cdot -0.5}{s}\right)}{\frac{\pi}{s} + 1} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
10. Add Preprocessing

Alternative 7: 24.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. *-commutative 25.1%

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

8. Simplified 25.1%

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

9. Final simplification 25.1%

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

Alternative 8: 24.9% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. neg-mul-1 25.1%

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

   3. log1p-define 25.1%

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

8. Simplified 25.1%

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

9. Final simplification 25.1%

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

Alternative 9: 13.0% accurate, 20.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in s around 0 25.0%

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

7. Taylor expanded in s around inf 13.0%

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

8. Final simplification 13.0%

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

Alternative 10: 11.5% accurate, 28.9× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \left(u \cdot \left(\pi \cdot -0.5 + 0.25 \cdot \left(\pi \cdot \frac{1}{u}\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* -4.0 (* u (+ (* PI -0.5) (* 0.25 (* PI (/ 1.0 u)))))))

C

float code(float u, float s) {
	return -4.0f * (u * ((((float) M_PI) * -0.5f) + (0.25f * (((float) M_PI) * (1.0f / u)))));
}

Julia

function code(u, s)
	return Float32(Float32(-4.0) * Float32(u * Float32(Float32(Float32(pi) * Float32(-0.5)) + Float32(Float32(0.25) * Float32(Float32(pi) * Float32(Float32(1.0) / u))))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * (u * ((single(pi) * single(-0.5)) + (single(0.25) * (single(pi) * (single(1.0) / u)))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 11.5%

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

   1. associate--r+ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. metadata-eval 11.5%

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

   4. cancel-sign-sub-inv 11.5%

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

   5. associate-*r* 11.5%

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

   6. distribute-rgt-out 11.5%

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

   7. metadata-eval 11.5%

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

   8. *-commutative 11.5%

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

   9. associate-*l* 11.5%

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

7. Simplified 11.5%

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

8. Taylor expanded in u around inf 11.5%

   \[\leadsto -4 \cdot \color{blue}{\left(u \cdot \left(-0.5 \cdot \pi + 0.25 \cdot \frac{\pi}{u}\right)\right)} \]
9. Step-by-step derivation

   1. div-inv 11.5%

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

10. Applied egg-rr 11.5%

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

11. Final simplification 11.5%

    \[\leadsto -4 \cdot \left(u \cdot \left(\pi \cdot -0.5 + 0.25 \cdot \left(\pi \cdot \frac{1}{u}\right)\right)\right) \]
12. Add Preprocessing

Alternative 11: 11.5% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.5\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* -4.0 (* PI (+ 0.25 (* u -0.5)))))

C

float code(float u, float s) {
	return -4.0f * (((float) M_PI) * (0.25f + (u * -0.5f)));
}

Julia

function code(u, s)
	return Float32(Float32(-4.0) * Float32(Float32(pi) * Float32(Float32(0.25) + Float32(u * Float32(-0.5)))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * (single(pi) * (single(0.25) + (u * single(-0.5))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 11.5%

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

   1. associate--r+ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. metadata-eval 11.5%

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

   4. cancel-sign-sub-inv 11.5%

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

   5. associate-*r* 11.5%

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

   6. distribute-rgt-out 11.5%

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

   7. metadata-eval 11.5%

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

   8. *-commutative 11.5%

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

   9. associate-*l* 11.5%

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

7. Simplified 11.5%

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

8. Taylor expanded in u around 0 11.5%

   \[\leadsto -4 \cdot \color{blue}{\left(-0.5 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)} \]
9. Step-by-step derivation

   1. +-commutative 11.5%

      \[\leadsto -4 \cdot \color{blue}{\left(0.25 \cdot \pi + -0.5 \cdot \left(u \cdot \pi\right)\right)} \]

   2. associate-*r* 11.5%

      \[\leadsto -4 \cdot \left(0.25 \cdot \pi + \color{blue}{\left(-0.5 \cdot u\right) \cdot \pi}\right) \]

   3. distribute-rgt-out 11.5%

      \[\leadsto -4 \cdot \color{blue}{\left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)} \]

10. Simplified 11.5%

    \[\leadsto -4 \cdot \color{blue}{\left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)} \]

11. Final simplification 11.5%

    \[\leadsto -4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.5\right)\right) \]
12. Add Preprocessing

Alternative 12: 11.5% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 11.5%

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

   1. associate--r+ 11.5%

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

   2. cancel-sign-sub-inv 11.5%

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

   3. metadata-eval 11.5%

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

   4. cancel-sign-sub-inv 11.5%

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

   5. associate-*r* 11.5%

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

   6. distribute-rgt-out 11.5%

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

   7. metadata-eval 11.5%

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

   8. *-commutative 11.5%

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

   9. associate-*l* 11.5%

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

7. Simplified 11.5%

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

8. Taylor expanded in u around inf 11.5%

   \[\leadsto -4 \cdot \color{blue}{\left(u \cdot \left(-0.5 \cdot \pi + 0.25 \cdot \frac{\pi}{u}\right)\right)} \]

9. Taylor expanded in u around 0 11.5%

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

    1. +-commutative 11.5%

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

    2. mul-1-neg 11.5%

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

    3. unsub-neg 11.5%

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

    4. *-commutative 11.5%

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

11. Simplified 11.5%

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

12. Final simplification 11.5%

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

Alternative 13: 11.2% accurate, 216.5× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 11.3%

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

   1. neg-mul-1 11.3%

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

7. Simplified 11.3%

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

Alternative 14: 10.4% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 0.0)

C

float code(float u, float s) {
	return 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(u, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 10.5%

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

6. Taylor expanded in s around 0 10.5%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(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))))