Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 12.8s

Alternatives: 9

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 1.0 (exp (* (/ 1.0 (sqrt s)) (/ PI (sqrt 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(((1.0f / sqrtf(s)) * (((float) M_PI) / sqrtf(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(pi) / Float32(-s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(Float32(1.0) / sqrt(s)) * Float32(Float32(pi) / sqrt(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(1.0) / sqrt(s)) * (single(pi) / sqrt(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%

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

   1. *-un-lft-identity 98.8%

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

   2. add-sqr-sqrt 98.8%

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

   3. times-frac 98.9%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 3: 25.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * ((u * single(2.0)) + (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%

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

4. Taylor expanded in s around -inf 25.1%

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

   1. +-commutative 25.1%

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

   2. fma-def 25.1%

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

6. Simplified 25.1%

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

7. Taylor expanded in u around 0 25.3%

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

   1. +-commutative 25.3%

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

   2. mul-1-neg 25.3%

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

   3. unsub-neg 25.3%

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

   4. associate-*r/ 25.3%

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

   5. *-commutative 25.3%

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

   6. +-commutative 25.3%

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

   7. log1p-def 25.3%

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

9. Simplified 25.3%

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

10. Taylor expanded in s around 0 25.4%

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

    1. mul-1-neg 25.4%

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

12. Simplified 25.4%

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

13. Final simplification 25.4%

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

Alternative 4: 25.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * ((u * single(2.0)) + log((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%

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

4. Taylor expanded in s around -inf 25.1%

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

   1. +-commutative 25.1%

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

   2. fma-def 25.1%

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

6. Simplified 25.1%

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

7. Taylor expanded in u around 0 25.3%

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

   1. +-commutative 25.3%

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

   2. mul-1-neg 25.3%

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

   3. unsub-neg 25.3%

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

   4. associate-*r/ 25.3%

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

   5. *-commutative 25.3%

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

   6. +-commutative 25.3%

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

   7. log1p-def 25.3%

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

9. Simplified 25.3%

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

10. Taylor expanded in s around 0 25.4%

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

    1. mul-1-neg 25.4%

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

12. Simplified 25.4%

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

    1. unsub-neg 25.4%

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

    2. log-div 25.3%

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

    3. clear-num 25.3%

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

    4. log-rec 25.4%

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

14. Applied egg-rr 25.4%

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

15. Final simplification 25.4%

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

Alternative 5: 25.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * log((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%

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

4. Taylor expanded in s around -inf 25.1%

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

   1. +-commutative 25.1%

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

   2. fma-def 25.1%

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

6. Simplified 25.1%

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

7. Taylor expanded in u around 0 25.3%

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

   1. +-commutative 25.3%

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

   2. mul-1-neg 25.3%

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

   3. unsub-neg 25.3%

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

   4. associate-*r/ 25.3%

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

   5. *-commutative 25.3%

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

   6. +-commutative 25.3%

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

   7. log1p-def 25.3%

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

9. Simplified 25.3%

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

10. Taylor expanded in s around 0 25.4%

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

    1. mul-1-neg 25.4%

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

12. Simplified 25.4%

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

13. Taylor expanded in u around 0 25.4%

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

    1. log-div 25.4%

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

15. Simplified 25.4%

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

16. Final simplification 25.4%

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

Alternative 6: 11.7% 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 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%

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

4. Taylor expanded in s around -inf 25.1%

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

   1. +-commutative 25.1%

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

   2. fma-def 25.1%

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

6. Simplified 25.1%

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

7. Taylor expanded in u around 0 25.3%

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

   1. +-commutative 25.3%

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

   2. mul-1-neg 25.3%

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

   3. unsub-neg 25.3%

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

   4. associate-*r/ 25.3%

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

   5. *-commutative 25.3%

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

   6. +-commutative 25.3%

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

   7. log1p-def 25.3%

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

9. Simplified 25.3%

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

10. Taylor expanded in s around inf 10.8%

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

    1. sub-neg 10.8%

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

    2. associate-*r* 10.8%

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

    3. neg-mul-1 10.8%

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

    4. distribute-rgt-out 10.8%

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

12. Simplified 10.8%

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

13. Final simplification 10.8%

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

Alternative 7: 14.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 2.799999911143463 \cdot 10^{-20}:\\ \;\;\;\;s \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-\pi\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 2.799999911143463e-20) (* s 0.0) (- PI)))

C

float code(float u, float s) {
	float tmp;
	if (s <= 2.799999911143463e-20f) {
		tmp = s * 0.0f;
	} else {
		tmp = -((float) M_PI);
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(2.799999911143463e-20))
		tmp = Float32(s * Float32(0.0));
	else
		tmp = Float32(-Float32(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	tmp = single(0.0);
	if (s <= single(2.799999911143463e-20))
		tmp = s * single(0.0);
	else
		tmp = -single(pi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 2.79999991e-20

   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%

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

   4. Taylor expanded in s around inf 13.5%

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

   if 2.79999991e-20 < s

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in u around 0 14.0%

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

      1. neg-mul-1 14.0%

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

   6. Simplified 14.0%

      \[\leadsto \color{blue}{-\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.799999911143463 \cdot 10^{-20}:\\ \;\;\;\;s \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-\pi\\ \end{array} \]

Alternative 8: 11.5% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in u around 0 10.6%

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

   1. neg-mul-1 10.6%

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

6. Simplified 10.6%

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

7. Final simplification 10.6%

   \[\leadsto -\pi \]

Alternative 9: 8.9% accurate, 146.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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%

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

4. Taylor expanded in s around -inf 25.1%

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

   1. +-commutative 25.1%

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

   2. fma-def 25.1%

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

6. Simplified 25.1%

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

7. Taylor expanded in u around 0 25.3%

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

   1. +-commutative 25.3%

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

   2. mul-1-neg 25.3%

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

   3. unsub-neg 25.3%

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

   4. associate-*r/ 25.3%

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

   5. *-commutative 25.3%

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

   6. +-commutative 25.3%

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

   7. log1p-def 25.3%

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

9. Simplified 25.3%

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

10. Taylor expanded in s around 0 25.4%

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

    1. mul-1-neg 25.4%

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

12. Simplified 25.4%

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

13. Taylor expanded in u around inf 9.2%

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

14. Final simplification 9.2%

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

Reproduce

herbie shell --seed 2023333 
(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))))