Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 19.6s

Alternatives: 10

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.4× speedup

Math

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

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(Float32(-s) * 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 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 37.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(1 - e^{\mathsf{log1p}\left(\log \left(-1 + \frac{1}{u \cdot 0.5 + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right)}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   1.0
   (exp
    (log1p
     (log
      (+ -1.0 (/ 1.0 (+ (* u 0.5) (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))))))))))))

C

float code(float u, float s) {
	return s * (1.0f - expf(log1pf(logf((-1.0f + (1.0f / ((u * 0.5f) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s)))))))))));
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(1.0) - exp(log1p(log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u * Float32(0.5)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))))))))))
end

Derivation

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%

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

4. Taylor expanded in s around inf 37.9%

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

   1. expm1-log1p-u 37.9%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + 1} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)\right)}\right) \]

   2. expm1-udef 37.9%

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

6. Applied egg-rr 37.9%

   \[\leadsto s \cdot \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(-1 + \frac{1}{u \cdot 0.5 - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}}\right)\right)} - 1\right)}\right) \]

7. Final simplification 37.9%

   \[\leadsto s \cdot \left(1 - e^{\mathsf{log1p}\left(\log \left(-1 + \frac{1}{u \cdot 0.5 + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right)}\right) \]

Alternative 3: 37.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{2} + \frac{1 - u}{1 + {e}^{\left(\frac{\pi}{s}\right)}}}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log (+ -1.0 (/ 1.0 (+ (/ u 2.0) (/ (- 1.0 u) (+ 1.0 (pow E (/ PI s))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + (1.0f / ((u / 2.0f) + ((1.0f - u) / (1.0f + powf(((float) M_E), (((float) M_PI) / s))))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u / Float32(2.0)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(pi) / s)))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (single(1.0) / ((u / single(2.0)) + ((single(1.0) - u) / (single(1.0) + (single(2.71828182845904523536) ^ (single(pi) / s))))))));
end

Derivation

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%

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

4. Taylor expanded in s around inf 37.9%

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

   1. *-un-lft-identity 37.9%

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

   2. exp-prod 37.9%

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

6. Applied egg-rr 37.9%

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

   1. exp-1-e 37.9%

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

8. Simplified 37.9%

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

9. Final simplification 37.9%

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

Alternative 4: 37.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(1 + \left(-1 - \log \left(-1 + \frac{1}{u \cdot 0.5 + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (+
   1.0
   (-
    -1.0
    (log
     (+ -1.0 (/ 1.0 (+ (* u 0.5) (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))))))))))

C

float code(float u, float s) {
	return s * (1.0f + (-1.0f - logf((-1.0f + (1.0f / ((u * 0.5f) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s))))))))));
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(1.0) + Float32(Float32(-1.0) - log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u * Float32(0.5)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * (single(1.0) + (single(-1.0) - log((single(-1.0) + (single(1.0) / ((u * single(0.5)) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s))))))))));
end

Derivation

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%

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

4. Taylor expanded in s around inf 37.9%

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

   1. expm1-log1p-u 37.9%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + 1} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)\right)}\right) \]

   2. expm1-udef 37.9%

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

6. Applied egg-rr 37.9%

   \[\leadsto s \cdot \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(-1 + \frac{1}{u \cdot 0.5 - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}}\right)\right)} - 1\right)}\right) \]

8. Applied egg-rr 37.9%

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

9. Final simplification 37.9%

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

Alternative 5: 37.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{2} + \frac{1 - u}{1 + e^{\pi \cdot \frac{1}{s}}}}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/ 1.0 (+ (/ u 2.0) (/ (- 1.0 u) (+ 1.0 (exp (* PI (/ 1.0 s)))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + (1.0f / ((u / 2.0f) + ((1.0f - u) / (1.0f + expf((((float) M_PI) * (1.0f / s)))))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u / Float32(2.0)) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) * Float32(Float32(1.0) / s))))))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (single(1.0) / ((u / single(2.0)) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) * (single(1.0) / s)))))))));
end

Derivation

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%

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

4. Taylor expanded in s around inf 37.9%

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

   1. clear-num 37.9%

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

   2. associate-/r/ 37.9%

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

6. Applied egg-rr 37.9%

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

7. Final simplification 37.9%

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

Alternative 6: 37.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{u \cdot 0.5 + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log (+ -1.0 (/ 1.0 (+ (* u 0.5) (/ (- 1.0 u) (+ 1.0 (exp (/ PI s))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + (1.0f / ((u * 0.5f) + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s))))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u * Float32(0.5)) + 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(0.5)) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) / s))))))));
end

Derivation

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%

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

4. Taylor expanded in s around inf 37.9%

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

   1. distribute-rgt-neg-out 37.9%

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

   2. add-sqr-sqrt 37.9%

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

   3. sqrt-unprod 37.9%

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

   4. sqr-neg 37.9%

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

   5. sqrt-unprod -0.0%

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

6. Applied egg-rr 37.9%

   \[\leadsto \color{blue}{-s \cdot \log \left(-1 + \frac{1}{u \cdot 0.5 - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}}\right)} \]
7. Step-by-step derivation

   1. distribute-lft-neg-in 37.9%

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

   2. *-commutative 37.9%

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

   3. +-commutative 37.9%

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

8. Simplified 37.9%

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

9. Final simplification 37.9%

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

Alternative 7: 11.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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%

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

4. Taylor expanded in s around inf 12.0%

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

   1. associate--r+ 12.0%

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

   2. cancel-sign-sub-inv 12.0%

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

   3. distribute-rgt-out-- 12.0%

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

   4. *-commutative 12.0%

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

   5. metadata-eval 12.0%

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

   6. metadata-eval 12.0%

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

   7. *-commutative 12.0%

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

6. Simplified 12.0%

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

7. Taylor expanded in u around 0 12.0%

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

   1. +-commutative 12.0%

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

   2. associate-*r* 12.0%

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

   3. *-commutative 12.0%

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

   4. distribute-rgt-out 12.0%

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

   5. *-commutative 12.0%

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

9. Simplified 12.0%

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

10. Final simplification 12.0%

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

Alternative 8: 37.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(-1 + \frac{2}{u}\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s (- (log (+ -1.0 (/ 2.0 u))))))

C

float code(float u, float s) {
	return s * -logf((-1.0f + (2.0f / u)));
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * -log(((-1.0e0) + (2.0e0 / u)))
end function

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(-1.0) + Float32(Float32(2.0) / u)))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(2.0) / u)));
end

Derivation

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%

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

4. Taylor expanded in s around inf 37.9%

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

5. Taylor expanded in s around inf 36.2%

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

6. Taylor expanded in s around 0 36.9%

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

   1. associate-*r* 36.9%

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

   2. neg-mul-1 36.9%

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

   3. sub-neg 36.9%

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

   4. associate-*r/ 36.9%

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

   5. metadata-eval 36.9%

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

   6. metadata-eval 36.9%

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

8. Simplified 36.9%

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

9. Final simplification 36.9%

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

Alternative 9: 11.4% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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%

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

4. Taylor expanded in u around 0 11.8%

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

   1. neg-mul-1 11.8%

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

6. Simplified 11.8%

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

7. Final simplification 11.8%

   \[\leadsto -\pi \]

Alternative 10: 10.3% accurate, 243.3× speedup

Math

\[\begin{array}{l} \\ s \cdot 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s 0.0))

C

float code(float u, float s) {
	return s * 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * 0.0e0
end function

Julia

function code(u, s)
	return Float32(s * Float32(0.0))
end

MATLAB

function tmp = code(u, s)
	tmp = s * single(0.0);
end

Derivation

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%

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

4. Taylor expanded in s around inf 37.9%

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

   1. expm1-log1p-u 37.9%

      \[\leadsto s \cdot \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{\frac{u}{1 + 1} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)\right)}\right) \]

   2. expm1-udef 37.9%

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

6. Applied egg-rr 37.9%

   \[\leadsto s \cdot \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(-1 + \frac{1}{u \cdot 0.5 - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}}\right)\right)} - 1\right)}\right) \]

7. Taylor expanded in s around inf 10.5%

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

8. Final simplification 10.5%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023305 
(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))))