Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 15.4s

Alternatives: 5

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

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+ (/ u (+ 1.0 (exp (/ PI (- s))))) (/ (- 1.0 u) (+ 1.0 (exp (/ PI 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(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) / 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 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 12.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.5%

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

   1. associate-*r/ 10.5%

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

   2. *-commutative 10.5%

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

   3. associate-/l* 10.5%

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

   4. associate--r+ 10.5%

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

   5. cancel-sign-sub-inv 10.5%

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

   6. distribute-rgt-out-- 10.5%

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

   7. *-commutative 10.5%

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

   8. metadata-eval 10.5%

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

   9. metadata-eval 10.5%

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

   10. *-commutative 10.5%

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

7. Simplified 10.5%

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

   1. add-log-exp 10.5%

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

   2. exp-prod 12.3%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 9.9%

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

   5. sqr-neg 9.9%

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

   6. sqrt-unprod 9.9%

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

   7. add-sqr-sqrt 9.9%

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

9. Applied egg-rr 12.1%

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

10. Taylor expanded in u around inf 12.3%

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

    1. associate-*r/ 12.3%

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

    2. *-commutative 12.3%

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

    3. *-commutative 12.3%

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

    4. associate-/l* 12.3%

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

12. Simplified 12.3%

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

13. Final simplification 12.3%

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

Alternative 3: 12.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-2, \pi \cdot \left(s \cdot u\right), \frac{0}{s}\right)}{s} \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (/ (fma -2.0 (* PI (* s u)) (/ 0.0 s)) s))

C

float code(float u, float s) {
	return fmaf(-2.0f, (((float) M_PI) * (s * u)), (0.0f / s)) / s;
}

Julia

function code(u, s)
	return Float32(fma(Float32(-2.0), Float32(Float32(pi) * Float32(s * u)), Float32(Float32(0.0) / s)) / s)
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.5%

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

   1. associate-*r/ 10.5%

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

   2. *-commutative 10.5%

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

   3. associate-/l* 10.5%

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

   4. associate--r+ 10.5%

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

   5. cancel-sign-sub-inv 10.5%

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

   6. distribute-rgt-out-- 10.5%

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

   7. *-commutative 10.5%

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

   8. metadata-eval 10.5%

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

   9. metadata-eval 10.5%

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

   10. *-commutative 10.5%

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

7. Simplified 10.5%

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

   1. add-log-exp 10.5%

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

   2. exp-prod 12.3%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 9.9%

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

   5. sqr-neg 9.9%

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

   6. sqrt-unprod 9.9%

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

   7. add-sqr-sqrt 9.9%

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

9. Applied egg-rr 12.1%

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

10. Taylor expanded in u around inf 12.3%

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

    1. associate-*r/ 12.3%

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

    2. *-commutative 12.3%

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

    3. *-commutative 12.3%

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

    4. associate-/l* 12.3%

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

12. Simplified 12.3%

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

13. Taylor expanded in s around inf 12.2%

    \[\leadsto \color{blue}{\frac{-2 \cdot \left(s \cdot \left(u \cdot \pi\right)\right) + 0.5 \cdot \frac{-4 \cdot \left({s}^{2} \cdot \left({u}^{2} \cdot {\pi}^{2}\right)\right) + 4 \cdot \left({s}^{2} \cdot \left({u}^{2} \cdot {\pi}^{2}\right)\right)}{s}}{s}} \]
14. Step-by-step derivation

15. Simplified 12.3%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \pi \cdot \left(u \cdot s\right), \frac{0}{s}\right)}{s}} \]

16. Final simplification 12.3%

    \[\leadsto \frac{\mathsf{fma}\left(-2, \pi \cdot \left(s \cdot u\right), \frac{0}{s}\right)}{s} \]
17. Add Preprocessing

Alternative 4: 12.3% accurate, 86.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.5%

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

   1. associate-*r/ 10.5%

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

   2. *-commutative 10.5%

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

   3. associate-/l* 10.5%

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

   4. associate--r+ 10.5%

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

   5. cancel-sign-sub-inv 10.5%

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

   6. distribute-rgt-out-- 10.5%

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

   7. *-commutative 10.5%

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

   8. metadata-eval 10.5%

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

   9. metadata-eval 10.5%

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

   10. *-commutative 10.5%

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

7. Simplified 10.5%

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

   1. add-log-exp 10.5%

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

   2. exp-prod 12.3%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 9.9%

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

   5. sqr-neg 9.9%

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

   6. sqrt-unprod 9.9%

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

   7. add-sqr-sqrt 9.9%

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

9. Applied egg-rr 12.1%

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

10. Taylor expanded in u around inf 12.3%

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

    1. associate-*r/ 12.3%

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

    2. *-commutative 12.3%

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

    3. *-commutative 12.3%

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

    4. associate-/l* 12.3%

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

12. Simplified 12.3%

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

13. Taylor expanded in s around 0 12.1%

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

    1. *-commutative 12.1%

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

    2. *-commutative 12.1%

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

    3. associate-*l* 12.1%

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

15. Simplified 12.1%

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

16. Final simplification 12.1%

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

Alternative 5: 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 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u around 0 10.4%

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

   1. neg-mul-1 10.4%

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

7. Simplified 10.4%

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

8. Final simplification 10.4%

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

Reproduce

herbie shell --seed 2024081 
(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))))