Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 13.7s

Alternatives: 6

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 25.2% accurate, 1.8× 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 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-def 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

6. Simplified 24.8%

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

7. Taylor expanded in u around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]
8. 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-def 25.1%

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

9. Simplified 25.1%

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

10. Taylor expanded in s around 0 25.2%

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

    1. associate-+r+ 25.2%

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

    2. +-commutative 25.2%

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

    3. mul-1-neg 25.2%

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

    4. unsub-neg 25.2%

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

12. Simplified 25.2%

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

13. Final simplification 25.2%

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

Alternative 3: 25.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-def 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

6. Simplified 24.8%

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

7. Taylor expanded in u around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]
8. 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-def 25.1%

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

9. Simplified 25.1%

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

10. Taylor expanded in s around 0 25.2%

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

    1. +-commutative 25.2%

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

    2. mul-1-neg 25.2%

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

    3. unsub-neg 25.2%

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

12. Simplified 25.2%

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

13. Final simplification 25.2%

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

Alternative 4: 25.1% accurate, 3.6× speedup

Math

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

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(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.8%

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

   1. +-commutative 24.8%

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

   2. fma-def 24.8%

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

   3. associate--r+ 24.8%

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

   4. cancel-sign-sub-inv 24.8%

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

   5. distribute-rgt-out-- 24.8%

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

   6. *-commutative 24.8%

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

   7. metadata-eval 24.8%

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

   8. metadata-eval 24.8%

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

   9. *-commutative 24.8%

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

6. Simplified 24.8%

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

7. Taylor expanded in u around 0 25.1%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \log \left(1 + \frac{\pi}{s}\right)\right)} \]
8. 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-def 25.1%

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

9. Simplified 25.1%

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

10. Final simplification 25.1%

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

Alternative 5: 11.6% accurate, 6.9× 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 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u around 0 2.4%

   \[\leadsto \color{blue}{-0.5 \cdot \left(s \cdot \left(\left(-1 \cdot \frac{{\left(e^{\frac{\pi}{s}} + 1\right)}^{2} \cdot \left({\left(1 + e^{\frac{\pi}{s}}\right)}^{2} \cdot {\left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{e^{\frac{\pi}{s}} + 1}\right)}^{2}\right)}{{\left(e^{\frac{\pi}{s}}\right)}^{2}} + 2 \cdot \frac{{\left(e^{\frac{\pi}{s}} + 1\right)}^{2} \cdot \left(\left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) \cdot \left(\left(1 + e^{\frac{\pi}{s}}\right) \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right)\right)}{e^{\frac{\pi}{s}}}\right) \cdot {u}^{2}\right)\right) + \left(-1 \cdot \pi + \frac{\left(e^{\frac{\pi}{s}} + 1\right) \cdot \left(s \cdot \left(\left(1 + e^{\frac{\pi}{s}}\right) \cdot \left(\left(\frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{e^{\frac{\pi}{s}} + 1}\right) \cdot u\right)\right)\right)}{e^{\frac{\pi}{s}}}\right)} \]

6. Simplified 2.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot s, \left(\frac{2 \cdot \left({\left(1 + e^{\frac{\pi}{s}}\right)}^{3} \cdot {\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)}^{2}\right)}{e^{\frac{\pi}{s}}} - \frac{{\left(\frac{1}{1 + e^{-\frac{\pi}{s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)}^{2} \cdot {\left(1 + e^{\frac{\pi}{s}}\right)}^{4}}{{\left(e^{\frac{\pi}{s}}\right)}^{2}}\right) \cdot \left(u \cdot u\right), \frac{s}{\frac{e^{\frac{\pi}{s}}}{\left(u \cdot \left(\frac{1}{1 + e^{-\frac{\pi}{s}}} + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)\right) \cdot {\left(1 + e^{\frac{\pi}{s}}\right)}^{2}}} - \pi\right)} \]

7. Taylor expanded in s around -inf 12.2%

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

   1. associate-*r* 12.2%

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

   2. fma-neg 12.2%

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

   3. distribute-rgt-out-- 12.2%

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

   4. metadata-eval 12.2%

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

9. Simplified 12.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-4 \cdot \left(\pi \cdot -0.5\right), u, -\pi\right)} \]
10. Step-by-step derivation

    1. expm1-log1p-u 12.2%

       \[\leadsto \mathsf{fma}\left(-4 \cdot \left(\pi \cdot -0.5\right), u, -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi\right)\right)}\right) \]

11. Applied egg-rr 12.2%

    \[\leadsto \mathsf{fma}\left(-4 \cdot \left(\pi \cdot -0.5\right), u, -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi\right)\right)}\right) \]

12. Taylor expanded in u around 0 12.2%

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

    1. +-commutative 12.2%

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

    2. associate-*r* 12.2%

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

    3. distribute-rgt-out 12.2%

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

14. Simplified 12.2%

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

15. Final simplification 12.2%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]

Alternative 6: 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 99.0%

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

   1. distribute-lft-neg-out 99.0%

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

   2. distribute-rgt-neg-in 99.0%

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

   3. sub-neg 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u around 0 12.0%

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

   1. mul-1-neg 12.0%

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

6. Simplified 12.0%

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

7. Final simplification 12.0%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023250 
(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))))