Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 15.2s

Alternatives: 9

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ PI (- s)))))
      (/ (- 1.0 u) (+ 1.0 (exp (* PI (/ 1.0 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) * (1.0f / 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(pi) / Float32(-s))))) + Float32(Float32(Float32(1.0) - u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) * Float32(Float32(1.0) / 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) * (single(1.0) / s))))))) + single(-1.0)));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. clear-num 98.8%

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

   2. associate-/r/ 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 2: 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 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 3: 25.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\log s - \left(-1 + e^{\mathsf{log1p}\left(\log \pi\right)}\right)\right) - -2 \cdot \left(s \cdot u\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (- (* s (- (log s) (+ -1.0 (exp (log1p (log PI)))))) (* -2.0 (* s u))))

C

float code(float u, float s) {
	return (s * (logf(s) - (-1.0f + expf(log1pf(logf(((float) M_PI))))))) - (-2.0f * (s * u));
}

Julia

function code(u, s)
	return Float32(Float32(s * Float32(log(s) - Float32(Float32(-1.0) + exp(log1p(log(Float32(pi))))))) - Float32(Float32(-2.0) * Float32(s * u)))
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-def 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in s around 0 24.5%

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

9. Taylor expanded in u around 0 24.8%

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

    1. expm1-log1p-u 24.8%

       \[\leadsto -1 \cdot \left(-2 \cdot \left(s \cdot u\right) + s \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \pi\right)\right)} + -1 \cdot \log s\right)\right) \]

    2. expm1-udef 24.8%

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

11. Applied egg-rr 24.8%

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

12. Final simplification 24.8%

    \[\leadsto s \cdot \left(\log s - \left(-1 + e^{\mathsf{log1p}\left(\log \pi\right)}\right)\right) - -2 \cdot \left(s \cdot u\right) \]
13. Add Preprocessing

Alternative 4: 25.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-def 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in s around 0 24.5%

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

9. Taylor expanded in u around 0 24.8%

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

10. Taylor expanded in s around 0 24.8%

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

11. Final simplification 24.8%

    \[\leadsto s \cdot \left(\left(\log s - u \cdot -2\right) - \log \pi\right) \]
12. Add Preprocessing

Alternative 5: 25.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

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. Add Preprocessing

5. Taylor expanded in s around -inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-def 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in s around 0 24.5%

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

9. Taylor expanded in u around 0 24.8%

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

10. Taylor expanded in s around 0 24.8%

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

    1. neg-mul-1 24.8%

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

    2. log-rec 24.8%

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

    3. +-commutative 24.8%

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

    4. associate-+r+ 24.8%

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

    5. log-rec 24.8%

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

    6. sub-neg 24.8%

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

    7. log-div 24.8%

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

    8. *-commutative 24.8%

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

12. Simplified 24.8%

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

13. Final simplification 24.8%

    \[\leadsto s \cdot \left(u \cdot \left(--2\right) - \log \left(\frac{\pi}{s}\right)\right) \]
14. Add Preprocessing

Alternative 6: 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%

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

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. Add Preprocessing

5. Taylor expanded in s around -inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-def 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in u around 0 24.7%

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

   1. log1p-def 24.7%

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

   2. associate-*r* 24.7%

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

   3. neg-mul-1 24.7%

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

10. Simplified 24.7%

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

11. Final simplification 24.7%

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

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 98.8%

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

3. Simplified 98.8%

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

   1. clear-num 98.8%

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

   2. associate-/r/ 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in s around inf 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. *-commutative 11.8%

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

   4. *-commutative 11.8%

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

   5. *-commutative 11.8%

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

   6. *-commutative 11.8%

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

   7. distribute-lft-out-- 11.8%

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

   8. metadata-eval 11.8%

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

   9. *-commutative 11.8%

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

   10. *-commutative 11.8%

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

   11. metadata-eval 11.8%

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

   12. associate-*r* 11.8%

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

   13. *-commutative 11.8%

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

   14. distribute-rgt-out 11.8%

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

9. Simplified 11.8%

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

10. Final simplification 11.8%

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

Alternative 8: 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 11.5%

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

   1. neg-mul-1 11.5%

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

7. Simplified 11.5%

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

8. Final simplification 11.5%

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

Alternative 9: 9.0% accurate, 121.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-def 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in s around 0 24.5%

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

9. Taylor expanded in u around 0 24.8%

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

10. Taylor expanded in u around inf 9.4%

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

    1. *-commutative 9.4%

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

12. Simplified 9.4%

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

13. Final simplification 9.4%

    \[\leadsto -2 \cdot \left(s \cdot \left(-u\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))))