Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 14.9s

Alternatives: 9

Speedup: 1.3×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t\_0\right) + t\_0} - 1\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

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

   1. clear-num 98.9%

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

   2. inv-pow 98.9%

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

6. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

8. Applied egg-rr 98.9%

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

9. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 1.3× 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%

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

5. Final simplification 98.9%

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

Alternative 3: 25.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.3%

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

   1. associate-*r/ 25.3%

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

   2. *-commutative 25.3%

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

   3. associate-/l* 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in s around 0 25.4%

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

    1. mul-1-neg 25.4%

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

    2. unsub-neg 25.4%

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

    3. distribute-rgt-in 25.4%

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

    4. *-commutative 25.4%

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

    5. +-commutative 25.4%

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

    6. mul-1-neg 25.4%

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

    7. unsub-neg 25.4%

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

    8. *-commutative 25.4%

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

    9. mul-1-neg 25.4%

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

    10. mul-1-neg 25.4%

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

    11. remove-double-neg 25.4%

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

11. Simplified 25.4%

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

12. Final simplification 25.4%

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

Alternative 4: 25.2% accurate, 2.1× 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.3%

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

   1. associate-*r/ 25.3%

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

   2. *-commutative 25.3%

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

   3. associate-/l* 25.3%

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

8. Simplified 25.3%

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

9. Taylor expanded in s around 0 25.4%

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

    1. mul-1-neg 25.4%

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

    2. distribute-rgt-neg-in 25.4%

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

    3. +-commutative 25.4%

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

    4. distribute-neg-in 25.4%

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

    5. mul-1-neg 25.4%

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

    6. *-commutative 25.4%

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

    7. unsub-neg 25.4%

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

    8. *-commutative 25.4%

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

    9. mul-1-neg 25.4%

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

    10. mul-1-neg 25.4%

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

    11. remove-double-neg 25.4%

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

11. Simplified 25.4%

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

12. Final simplification 25.4%

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

Alternative 5: 25.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(1 + 4 \cdot \left(\pi \cdot \frac{0.25}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (+ 1.0 (* 4.0 (* PI (/ 0.25 s)))))))

C

float code(float u, float s) {
	return -s * logf((1.0f + (4.0f * (((float) M_PI) * (0.25f / s)))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(1.0) + Float32(Float32(4.0) * Float32(Float32(pi) * Float32(Float32(0.25) / s))))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(1.0) + (single(4.0) * (single(pi) * (single(0.25) / 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%

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

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.3%

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

   1. associate-*r/ 25.3%

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

   2. *-commutative 25.3%

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

   3. associate-/l* 25.3%

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

8. Simplified 25.3%

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

9. Final simplification 25.3%

   \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \left(\pi \cdot \frac{0.25}{s}\right)\right) \]
10. Add Preprocessing

Alternative 6: 25.1% accurate, 4.1× speedup

Math

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

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(Float32(-s) * log1p(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%

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

5. Taylor expanded in s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.3%

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

   1. associate-*r/ 25.3%

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

   2. *-commutative 25.3%

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

   3. associate-/l* 25.3%

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

8. Simplified 25.3%

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

   1. associate-*r/ 25.3%

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

   2. associate-*l/ 25.3%

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

   3. *-un-lft-identity 25.3%

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

   4. log-prod 25.3%

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

   5. metadata-eval 25.3%

      \[\leadsto \left(-s\right) \cdot \left(\color{blue}{0} + \log \left(1 + 4 \cdot \left(\frac{\pi}{s} \cdot 0.25\right)\right)\right) \]

   6. log1p-define 25.3%

      \[\leadsto \left(-s\right) \cdot \left(0 + \color{blue}{\mathsf{log1p}\left(4 \cdot \left(\frac{\pi}{s} \cdot 0.25\right)\right)}\right) \]

   7. associate-*l/ 25.3%

      \[\leadsto \left(-s\right) \cdot \left(0 + \mathsf{log1p}\left(4 \cdot \color{blue}{\frac{\pi \cdot 0.25}{s}}\right)\right) \]

   8. associate-*r/ 25.3%

      \[\leadsto \left(-s\right) \cdot \left(0 + \mathsf{log1p}\left(4 \cdot \color{blue}{\left(\pi \cdot \frac{0.25}{s}\right)}\right)\right) \]

   9. associate-*r* 25.3%

      \[\leadsto \left(-s\right) \cdot \left(0 + \mathsf{log1p}\left(\color{blue}{\left(4 \cdot \pi\right) \cdot \frac{0.25}{s}}\right)\right) \]

10. Applied egg-rr 25.3%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\left(0 + \mathsf{log1p}\left(\left(4 \cdot \pi\right) \cdot \frac{0.25}{s}\right)\right)} \]
11. Step-by-step derivation

    1. +-lft-identity 25.3%

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

    2. associate-*r/ 25.3%

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

    3. *-commutative 25.3%

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

    4. associate-*l* 25.3%

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

    5. metadata-eval 25.3%

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

    6. *-rgt-identity 25.3%

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

12. Simplified 25.3%

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

13. Final simplification 25.3%

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

Alternative 7: 11.6% accurate, 30.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around inf 11.8%

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

6. Taylor expanded in u around -inf 11.8%

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

   1. mul-1-neg 11.8%

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

   2. distribute-rgt-neg-in 11.8%

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

   3. cancel-sign-sub-inv 11.8%

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

   4. +-commutative 11.8%

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

   5. *-commutative 11.8%

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

   6. metadata-eval 11.8%

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

   7. *-commutative 11.8%

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

   8. associate-+l+ 11.8%

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

   9. associate-*r/ 11.8%

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

   10. *-commutative 11.8%

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

   11. associate-/l* 11.8%

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

   12. distribute-lft-out 11.8%

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

   13. metadata-eval 11.8%

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

8. Simplified 11.8%

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

9. Final simplification 11.8%

   \[\leadsto 4 \cdot \left(u \cdot \left(\pi \cdot \frac{-0.25}{u} - \pi \cdot -0.5\right)\right) \]
10. Add Preprocessing

Alternative 8: 11.6% 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 11.8%

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

   1. associate--r+ 11.8%

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

   2. cancel-sign-sub-inv 11.8%

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

   3. metadata-eval 11.8%

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

   4. cancel-sign-sub-inv 11.8%

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

   5. associate-*r* 11.8%

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

   6. distribute-rgt-out 11.8%

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

   7. *-commutative 11.8%

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

   8. metadata-eval 11.8%

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

   9. *-commutative 11.8%

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

   10. associate-*l* 11.8%

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

7. Simplified 11.8%

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

8. Taylor expanded in u around 0 11.8%

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

   1. associate-*r* 11.8%

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

   2. distribute-rgt-out 11.8%

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

   3. *-commutative 11.8%

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

10. Simplified 11.8%

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

11. Final simplification 11.8%

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

Alternative 9: 11.4% 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.6%

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

   1. neg-mul-1 11.6%

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

7. Simplified 11.6%

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

8. Final simplification 11.6%

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

Reproduce

herbie shell --seed 2024085 
(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))))