Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 10.1s

Alternatives: 7

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, 1.4× 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 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 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%

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

Alternative 2: 25.2% accurate, 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 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 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 24.8%

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

5. Taylor expanded in u around 0 25.0%

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

6. Taylor expanded in s around 0 25.2%

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

7. Final simplification 25.2%

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

Alternative 3: 25.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 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 24.8%

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

5. Taylor expanded in u around 0 25.0%

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

6. Taylor expanded in s around 0 25.2%

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

   1. +-commutative 25.2%

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

   2. mul-1-neg 25.2%

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

   3. sub-neg 25.2%

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

   4. log-div 25.0%

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

8. Simplified 25.0%

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

9. Final simplification 25.0%

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

Alternative 4: 25.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 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 24.8%

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

5. Taylor expanded in u around 0 25.0%

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

   1. distribute-rgt-neg-out 25.0%

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

   2. +-commutative 25.0%

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

   3. log1p-udef 25.0%

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

   4. associate-*r* 25.0%

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

   5. metadata-eval 25.0%

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

   6. *-un-lft-identity 25.0%

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

7. Applied egg-rr 25.0%

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

   1. distribute-lft-neg-in 25.0%

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

9. Simplified 25.0%

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

10. Final simplification 25.0%

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

Alternative 5: 24.9% accurate, 3.6× speedup

Math

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

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%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 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 24.8%

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

5. Taylor expanded in u around 0 25.0%

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

6. Taylor expanded in s around 0 25.2%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 7.8%

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

   3. sqr-neg 7.8%

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

   4. sqrt-unprod 7.8%

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

   5. add-sqr-sqrt 7.8%

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

   6. +-commutative 7.8%

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

   7. distribute-lft-in 7.8%

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

   8. add-sqr-sqrt 7.8%

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

   9. sqrt-unprod 7.8%

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

   10. mul-1-neg 7.8%

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

   11. mul-1-neg 7.8%

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

   12. sqr-neg 7.8%

       \[\leadsto s \cdot \log \pi + s \cdot \sqrt{\color{blue}{\log s \cdot \log s}} \]

   13. sqrt-unprod -0.0%

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

   14. add-sqr-sqrt 25.0%

       \[\leadsto s \cdot \log \pi + s \cdot \color{blue}{\log s} \]

8. Applied egg-rr 25.0%

   \[\leadsto \color{blue}{s \cdot \log \pi + s \cdot \log s} \]
9. Step-by-step derivation

   1. +-commutative 25.0%

      \[\leadsto \color{blue}{s \cdot \log s + s \cdot \log \pi} \]

   2. distribute-lft-out 25.0%

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

   3. log-prod 25.0%

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

10. Simplified 25.0%

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

11. Final simplification 25.0%

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

Alternative 6: 13.2% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 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 24.8%

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

5. Taylor expanded in u around 0 25.0%

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

6. Taylor expanded in s around inf 11.0%

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

   1. distribute-rgt-neg-out 11.0%

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

   2. neg-sub0 11.0%

      \[\leadsto \color{blue}{0 - s \cdot \frac{\pi}{s}} \]

   3. div-inv 11.0%

      \[\leadsto 0 - s \cdot \color{blue}{\left(\pi \cdot \frac{1}{s}\right)} \]

   4. inv-pow 11.0%

      \[\leadsto 0 - s \cdot \left(\pi \cdot \color{blue}{{s}^{-1}}\right) \]

   5. exp-to-pow 11.0%

      \[\leadsto 0 - s \cdot \left(\pi \cdot \color{blue}{e^{\log s \cdot -1}}\right) \]

   6. *-commutative 11.0%

      \[\leadsto 0 - s \cdot \left(\pi \cdot e^{\color{blue}{-1 \cdot \log s}}\right) \]

   7. add-sqr-sqrt 11.0%

      \[\leadsto 0 - s \cdot \left(\pi \cdot e^{\color{blue}{\sqrt{-1 \cdot \log s} \cdot \sqrt{-1 \cdot \log s}}}\right) \]

   8. sqrt-unprod 11.0%

      \[\leadsto 0 - s \cdot \left(\pi \cdot e^{\color{blue}{\sqrt{\left(-1 \cdot \log s\right) \cdot \left(-1 \cdot \log s\right)}}}\right) \]

   9. mul-1-neg 11.0%

      \[\leadsto 0 - s \cdot \left(\pi \cdot e^{\sqrt{\color{blue}{\left(-\log s\right)} \cdot \left(-1 \cdot \log s\right)}}\right) \]

   10. mul-1-neg 11.0%

       \[\leadsto 0 - s \cdot \left(\pi \cdot e^{\sqrt{\left(-\log s\right) \cdot \color{blue}{\left(-\log s\right)}}}\right) \]

   11. sqr-neg 11.0%

       \[\leadsto 0 - s \cdot \left(\pi \cdot e^{\sqrt{\color{blue}{\log s \cdot \log s}}}\right) \]

   12. sqrt-unprod -0.0%

       \[\leadsto 0 - s \cdot \left(\pi \cdot e^{\color{blue}{\sqrt{\log s} \cdot \sqrt{\log s}}}\right) \]

   13. add-sqr-sqrt 13.2%

       \[\leadsto 0 - s \cdot \left(\pi \cdot e^{\color{blue}{\log s}}\right) \]

   14. add-exp-log 13.2%

       \[\leadsto 0 - s \cdot \left(\pi \cdot \color{blue}{s}\right) \]

8. Applied egg-rr 13.2%

   \[\leadsto \color{blue}{0 - s \cdot \left(\pi \cdot s\right)} \]
9. Step-by-step derivation

   1. neg-sub0 13.2%

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

   2. distribute-rgt-neg-in 13.2%

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

   3. *-commutative 13.2%

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

   4. distribute-rgt-neg-out 13.2%

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

10. Simplified 13.2%

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

11. Final simplification 13.2%

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

Alternative 7: 11.3% 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%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 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 11.0%

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

   1. mul-1-neg 11.0%

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

6. Simplified 11.0%

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

7. Final simplification 11.0%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023182 
(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))))