Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 15.6s

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

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

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(pi) / Float32(-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%

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

5. Final simplification 98.7%

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

Alternative 2: 25.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * (-log(single(4.0)) - log(((single(pi) * ((u * single(-0.25)) - single(-0.25))) / 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%

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

5. Taylor expanded in s around -inf 24.1%

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

6. Taylor expanded in u around 0 24.5%

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

7. Taylor expanded in s around 0 24.7%

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

   1. mul-1-neg 24.7%

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

   2. distribute-rgt-neg-in 24.7%

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

   3. log-prod 24.7%

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

   4. neg-mul-1 24.7%

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

   5. associate-+l+ 24.7%

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

   6. distribute-lft-out-- 24.7%

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

   7. *-commutative 24.7%

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

   8. fma-neg 24.7%

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

   9. log-rec 24.6%

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

   10. log-prod 24.6%

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

9. Simplified 24.6%

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

10. Final simplification 24.6%

    \[\leadsto s \cdot \left(\left(-\log 4\right) - \log \left(\frac{\pi \cdot \left(u \cdot -0.25 - -0.25\right)}{s}\right)\right) \]
11. 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%

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

5. Taylor expanded in s around -inf 24.1%

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

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. Step-by-step derivation

   1. mul-1-neg 24.5%

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

   2. *-commutative 24.5%

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

   3. distribute-rgt-neg-in 24.5%

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

   4. log1p-def 24.5%

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

8. Simplified 24.5%

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

9. Taylor expanded in s around 0 24.6%

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

    1. mul-1-neg 24.6%

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

    2. *-commutative 24.6%

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

    3. distribute-rgt-neg-in 24.6%

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

    4. mul-1-neg 24.6%

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

    5. unsub-neg 24.6%

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

11. Simplified 24.6%

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

12. Taylor expanded in s around 0 24.6%

    \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\log \pi - \log s\right)\right)} \]
13. 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. log-div 24.6%

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

    5. sub-neg 24.6%

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

    6. +-commutative 24.6%

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

    7. distribute-neg-in 24.6%

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

    8. remove-double-neg 24.6%

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

    9. sub-neg 24.6%

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

    10. log-div 24.6%

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

14. Simplified 24.6%

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

15. Final simplification 24.6%

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

Alternative 4: 11.5% accurate, 28.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} \]
4. 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. *-commutative 11.2%

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

   2. associate-*l* 11.2%

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

   3. +-commutative 11.2%

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

   4. associate-*r* 11.2%

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

   5. distribute-rgt-out 11.2%

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

   6. *-commutative 11.2%

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

7. Simplified 11.2%

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

8. Final simplification 11.2%

   \[\leadsto -4 \cdot \left(u \cdot \left(\pi \cdot -0.25\right) - \pi \cdot \left(-0.25 + u \cdot 0.25\right)\right) \]
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%

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

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)} \]
4. 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. associate-*r* 11.2%

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

   2. *-commutative 11.2%

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

   3. distribute-rgt-out 11.2%

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

10. Simplified 11.2%

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

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, 72.2× speedup

Math

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

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(-Float32(s * 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%

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

5. Taylor expanded in s around -inf 24.1%

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

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. Step-by-step derivation

   1. mul-1-neg 24.5%

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

   2. *-commutative 24.5%

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

   3. distribute-rgt-neg-in 24.5%

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

   4. log1p-def 24.5%

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

8. Simplified 24.5%

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

9. Taylor expanded in s around inf 10.8%

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

    1. associate-*l/ 10.8%

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

    2. frac-2neg 10.8%

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

    3. add-sqr-sqrt -0.0%

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

    4. sqrt-unprod 8.8%

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

    5. sqr-neg 8.8%

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

    6. sqrt-unprod 4.9%

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

    7. add-sqr-sqrt 4.9%

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

    8. add-sqr-sqrt -0.0%

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

    9. sqrt-unprod 9.1%

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

    10. sqr-neg 9.1%

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

    11. sqrt-unprod 10.8%

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

    12. add-sqr-sqrt 10.8%

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

11. Applied egg-rr 10.8%

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

12. Final simplification 10.8%

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

Alternative 8: 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%

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

Alternative 9: 4.6% accurate, 433.0× 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(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%

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

5. Taylor expanded in s around -inf 24.1%

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

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. Step-by-step derivation

   1. mul-1-neg 24.5%

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

   2. *-commutative 24.5%

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

   3. distribute-rgt-neg-in 24.5%

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

   4. log1p-def 24.5%

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

8. Simplified 24.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \left(-s\right)} \]
9. Step-by-step derivation

   1. add-sqr-sqrt -0.0%

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

   2. pow2 -0.0%

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

   3. *-commutative -0.0%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 9.3%

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

   6. sqr-neg 9.3%

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

   7. sqrt-unprod 7.9%

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

   8. add-sqr-sqrt 7.9%

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

10. Applied egg-rr 7.9%

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

11. Taylor expanded in s around inf 4.9%

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

12. Final simplification 4.9%

    \[\leadsto \pi \]
13. 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))))