Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 15.1s

Alternatives: 9

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

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

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(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + 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 98.7%

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

3. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 2: 25.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in s around -inf 24.1%

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

6. Taylor expanded in u around 0 24.5%

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

7. Taylor expanded in s around 0 24.6%

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

   1. neg-mul-1 24.6%

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

   2. associate-+r+ 24.6%

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

   3. sub-neg 24.6%

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

9. Simplified 24.6%

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

    1. diff-log 24.5%

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

    2. clear-num 24.5%

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

    3. log-rec 24.6%

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

11. Applied egg-rr 24.6%

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

12. Final simplification 24.6%

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

Alternative 3: 25.2% accurate, 4.1× 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 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in s around -inf 24.1%

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

6. Taylor expanded in s around 0 24.1%

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

8. Simplified 24.1%

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

9. Taylor expanded in u around 0 24.6%

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

10. Taylor expanded in u around 0 24.6%

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

    1. mul-1-neg 24.6%

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

    2. log-div 24.5%

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

    3. distribute-rgt-neg-in 24.5%

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

    4. mul-1-neg 24.5%

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

    5. log-div 24.6%

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

    6. sub-neg 24.6%

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

    7. remove-double-neg 24.6%

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

    8. sub-neg 24.6%

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

    9. mul-1-neg 24.6%

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

    10. log-rec 24.5%

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

    11. mul-1-neg 24.5%

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

    12. log-rec 24.6%

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

    13. mul-1-neg 24.6%

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

    14. remove-double-neg 24.6%

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

12. Simplified 24.6%

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

13. Final simplification 24.6%

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

Alternative 4: 11.5% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in s around -inf 11.2%

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

   1. associate--r+ 11.2%

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

   2. cancel-sign-sub-inv 11.2%

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

   3. metadata-eval 11.2%

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

   4. cancel-sign-sub-inv 11.2%

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

   5. associate-*r* 11.2%

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

   6. distribute-rgt-out 11.2%

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

   7. *-commutative 11.2%

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

   8. metadata-eval 11.2%

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

   9. *-commutative 11.2%

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

   10. associate-*l* 11.2%

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

7. Simplified 11.2%

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

8. Final simplification 11.2%

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

Alternative 5: 11.5% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in s around inf 11.2%

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

   1. associate--r+ 11.2%

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

   2. cancel-sign-sub-inv 11.2%

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

   3. distribute-rgt-out-- 11.2%

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

   4. *-commutative 11.2%

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

   5. metadata-eval 11.2%

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

   6. metadata-eval 11.2%

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

   7. *-commutative 11.2%

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

7. Simplified 11.2%

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

8. Final simplification 11.2%

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

Alternative 6: 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 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in s around inf 11.2%

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

   1. associate--r+ 11.2%

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

   2. cancel-sign-sub-inv 11.2%

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

   3. distribute-rgt-out-- 11.2%

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

   4. *-commutative 11.2%

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

   5. metadata-eval 11.2%

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

   6. metadata-eval 11.2%

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

   7. *-commutative 11.2%

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

7. Simplified 11.2%

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

8. Taylor expanded in u around 0 11.2%

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

   1. +-commutative 11.2%

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

   2. associate-*r* 11.2%

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

   3. *-commutative 11.2%

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

   4. distribute-rgt-out 11.2%

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

10. Simplified 11.2%

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

11. Final simplification 11.2%

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

Alternative 7: 11.3% accurate, 61.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{s}{s \cdot \pi}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in s around -inf 24.1%

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

6. Taylor expanded in u around 0 24.5%

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

7. Taylor expanded in s around inf 10.8%

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

   1. associate-*r/ 10.8%

      \[\leadsto -1 \cdot \color{blue}{\frac{s \cdot \pi}{s}} \]

   2. clear-num 10.8%

      \[\leadsto -1 \cdot \color{blue}{\frac{1}{\frac{s}{s \cdot \pi}}} \]

9. Applied egg-rr 10.8%

   \[\leadsto -1 \cdot \color{blue}{\frac{1}{\frac{s}{s \cdot \pi}}} \]

10. Final simplification 10.8%

    \[\leadsto \frac{-1}{\frac{s}{s \cdot \pi}} \]
11. Add Preprocessing

Alternative 8: 11.3% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in s around -inf 24.1%

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

6. Taylor expanded in u around 0 24.5%

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

7. Taylor expanded in s around inf 10.8%

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

   1. associate-*r/ 10.8%

      \[\leadsto -1 \cdot \color{blue}{\frac{s \cdot \pi}{s}} \]

9. Applied egg-rr 10.8%

   \[\leadsto -1 \cdot \color{blue}{\frac{s \cdot \pi}{s}} \]

10. Final simplification 10.8%

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

Alternative 9: 11.3% 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 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in u around 0 10.8%

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

   1. neg-mul-1 10.8%

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

7. Simplified 10.8%

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

8. Final simplification 10.8%

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

Reproduce

herbie shell --seed 2024041 
(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))))