Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 16.5s

Alternatives: 8

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.3× 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%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 25.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(log(s) + Float32(Float32(u * Float32(Float32(2.0) - Float32(u * Float32(-2.0)))) - log1p(Float32(Float32(pi) + Float32(-1.0))))))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 25.7%

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

6. Taylor expanded in s around 0 25.6%

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

8. Simplified 25.6%

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

9. Taylor expanded in u around 0 25.9%

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

    1. log1p-expm1-u 25.9%

       \[\leadsto \left(\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \pi\right)\right)} + u \cdot \left(-2 \cdot u - 2\right)\right) - \log s\right) \cdot \left(-s\right) \]

    2. expm1-undefine 25.9%

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

    3. add-exp-log 25.9%

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

11. Applied egg-rr 25.9%

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

12. Final simplification 25.9%

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

Alternative 3: 25.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 25.7%

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

6. Taylor expanded in s around 0 25.6%

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

8. Simplified 25.6%

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

9. Taylor expanded in u around 0 25.9%

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

10. Final simplification 25.9%

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

Alternative 4: 25.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(u * Float32(s * Float32(2.0))) - Float32(s * log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 25.7%

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

   1. +-commutative 25.7%

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

   2. fma-define 25.7%

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

   3. associate--r+ 25.7%

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

   4. cancel-sign-sub-inv 25.7%

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

   5. distribute-rgt-out-- 25.7%

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

   6. *-commutative 25.7%

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

   7. metadata-eval 25.7%

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

   8. metadata-eval 25.7%

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

   9. *-commutative 25.7%

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

7. Simplified 25.7%

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

8. Taylor expanded in u around 0 25.9%

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

   1. +-commutative 25.9%

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

   2. mul-1-neg 25.9%

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

   3. unsub-neg 25.9%

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

   4. *-commutative 25.9%

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

   5. associate-/l* 25.9%

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

   6. associate-*r* 25.9%

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

   7. *-commutative 25.9%

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

   8. associate-*r/ 25.9%

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

   9. log1p-define 25.9%

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

10. Simplified 25.9%

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

11. Taylor expanded in s around 0 25.9%

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

12. Final simplification 25.9%

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

Alternative 5: 25.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 25.7%

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

   1. distribute-lft-neg-out 25.7%

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

   2. neg-sub0 25.7%

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

   3. log1p-define 25.7%

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

7. Applied egg-rr 25.7%

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

   1. neg-sub0 25.7%

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

   2. *-commutative 25.7%

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

   3. distribute-rgt-neg-in 25.7%

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

9. Simplified 25.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{4 \cdot \left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)}{s}\right) \cdot \left(-s\right)} \]
10. Step-by-step derivation

    1. clear-num 25.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{s}{4 \cdot \left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)}}}\right) \cdot \left(-s\right) \]

    2. inv-pow 25.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{{\left(\frac{s}{4 \cdot \left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)}\right)}^{-1}}\right) \cdot \left(-s\right) \]

    3. *-un-lft-identity 25.7%

       \[\leadsto \mathsf{log1p}\left({\left(\frac{\color{blue}{1 \cdot s}}{4 \cdot \left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)}\right)}^{-1}\right) \cdot \left(-s\right) \]

    4. times-frac 25.7%

       \[\leadsto \mathsf{log1p}\left({\color{blue}{\left(\frac{1}{4} \cdot \frac{s}{\pi \cdot \left(0.25 + -0.5 \cdot u\right)}\right)}}^{-1}\right) \cdot \left(-s\right) \]

    5. metadata-eval 25.7%

       \[\leadsto \mathsf{log1p}\left({\left(\color{blue}{0.25} \cdot \frac{s}{\pi \cdot \left(0.25 + -0.5 \cdot u\right)}\right)}^{-1}\right) \cdot \left(-s\right) \]

    6. +-commutative 25.7%

       \[\leadsto \mathsf{log1p}\left({\left(0.25 \cdot \frac{s}{\pi \cdot \color{blue}{\left(-0.5 \cdot u + 0.25\right)}}\right)}^{-1}\right) \cdot \left(-s\right) \]

    7. fma-define 25.7%

       \[\leadsto \mathsf{log1p}\left({\left(0.25 \cdot \frac{s}{\pi \cdot \color{blue}{\mathsf{fma}\left(-0.5, u, 0.25\right)}}\right)}^{-1}\right) \cdot \left(-s\right) \]

11. Applied egg-rr 25.7%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{{\left(0.25 \cdot \frac{s}{\pi \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)}\right)}^{-1}}\right) \cdot \left(-s\right) \]
12. Step-by-step derivation

    1. unpow-1 25.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1}{0.25 \cdot \frac{s}{\pi \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)}}}\right) \cdot \left(-s\right) \]

    2. associate-*r/ 25.7%

       \[\leadsto \mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{0.25 \cdot s}{\pi \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)}}}\right) \cdot \left(-s\right) \]

    3. times-frac 25.7%

       \[\leadsto \mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{0.25}{\pi} \cdot \frac{s}{\mathsf{fma}\left(-0.5, u, 0.25\right)}}}\right) \cdot \left(-s\right) \]

13. Simplified 25.7%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{0.25}{\pi} \cdot \frac{s}{\mathsf{fma}\left(-0.5, u, 0.25\right)}}}\right) \cdot \left(-s\right) \]

14. Taylor expanded in u around 0 25.8%

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

15. Final simplification 25.8%

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

Alternative 6: 25.2% 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 25.7%

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

6. Taylor expanded in u around 0 25.8%

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

   1. mul-1-neg 25.8%

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

   2. *-commutative 25.8%

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

   3. distribute-rgt-neg-in 25.8%

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

   4. log1p-define 25.8%

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

8. Simplified 25.8%

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

9. Final simplification 25.8%

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

Alternative 7: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 25.7%

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

   1. +-commutative 25.7%

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

   2. fma-define 25.7%

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

   3. associate--r+ 25.7%

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

   4. cancel-sign-sub-inv 25.7%

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

   5. distribute-rgt-out-- 25.7%

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

   6. *-commutative 25.7%

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

   7. metadata-eval 25.7%

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

   8. metadata-eval 25.7%

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

   9. *-commutative 25.7%

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

7. Simplified 25.7%

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

8. Taylor expanded in s around inf 11.8%

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

   1. distribute-lft-in 11.8%

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

   2. associate-*r* 11.8%

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

   3. metadata-eval 11.8%

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

   4. neg-mul-1 11.8%

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

   5. *-commutative 11.8%

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

   6. associate-*r* 11.8%

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

   7. metadata-eval 11.8%

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

   8. *-commutative 11.8%

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

   9. associate-*r* 11.8%

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

10. Simplified 11.8%

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

11. Taylor expanded in u around 0 11.8%

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

    1. sub-neg 11.8%

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

    2. associate-*r* 11.8%

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

    3. mul-1-neg 11.8%

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

    4. distribute-rgt-out 11.8%

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

    5. metadata-eval 11.8%

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

    6. distribute-lft-neg-in 11.8%

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

    7. *-commutative 11.8%

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

    8. distribute-rgt-neg-in 11.8%

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

    9. metadata-eval 11.8%

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

13. Simplified 11.8%

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

14. Final simplification 11.8%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]
15. Add Preprocessing

Alternative 8: 11.4% accurate, 216.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 11.6%

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

   1. neg-mul-1 11.6%

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

7. Simplified 11.6%

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

8. Final simplification 11.6%

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

Reproduce

herbie shell --seed 2024060 
(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))))