Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 17.2s

Alternatives: 10

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(u / Float32(Float32(-1.0) - exp(Float32(Float32(pi) / s))))) + Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ 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)))) + (u / (single(-1.0) - exp((single(pi) / s))))) + (single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ (single(pi) / s)))))) + single(-1.0)));
end

Derivation

1. Initial program 99.1%

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

   1. lift-PI.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.1

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

4. Applied egg-rr 99.1%

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

   1. lift-/.f32 N/A

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

   2. rem-log-exp N/A

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

   3. unpow1 N/A

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

   4. lift-PI.f32 N/A

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

   5. pow-to-exp N/A

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

   6. pow-unpow N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. exp-prod N/A

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

   10. pow-pow N/A

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

   11. *-commutative N/A

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

   12. lift-/.f32 N/A

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

   13. div-inv N/A

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

   14. lift-/.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. exp-1-e N/A

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

   17. lower-E.f32 99.1

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

6. Applied egg-rr 99.1%

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

8. Applied egg-rr 99.1%

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

9. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. lift-PI.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 99.1

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

4. Applied egg-rr 99.1%

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

6. Applied egg-rr 99.1%

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

8. Applied egg-rr 99.1%

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

9. Final simplification 99.1%

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

Alternative 3: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

5. Simplified 99.1%

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

6. Taylor expanded in s around -inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}, u, u\right)}\right)} - 1\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)}\right)}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}, u, u\right)}\right)} - 1\right) \]

   2. unsub-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}, u, u\right)}\right)} - 1\right) \]

   3. lower--.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}, u, u\right)}\right)} - 1\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}}\right)}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}, u, u\right)}\right)} - 1\right) \]

8. Simplified 98.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{\left(-\pi\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, 0.5 \cdot \left(\pi \cdot \pi\right)\right)}{s}}{s}\right)}}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\pi}{s}}, u, u\right)}\right)} - 1\right) \]

9. Taylor expanded in s around 0

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

    1. +-commutative N/A

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

    2. *-rgt-identity N/A

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

    3. distribute-lft-in N/A

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

    4. lower-*.f32 N/A

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

    5. +-commutative N/A

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

    6. lower-+.f32 N/A

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

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 4: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf((-1.0f + (1.0f / (u * ((1.0f / (1.0f + expf((((float) M_PI) / -s)))) + (-1.0f / (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(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(-1.0) / 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) / (single(1.0) + exp((single(pi) / -s)))) + (single(-1.0) / (single(1.0) + exp((single(pi) / s)))))))));
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. mul-1-neg N/A

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

   10. lower-/.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f32 N/A

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

   14. distribute-neg-frac N/A

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

5. Simplified 97.4%

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

6. Final simplification 97.4%

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

Alternative 5: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{2 + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s \cdot s}, \mathsf{fma}\left(0.5, \frac{\pi \cdot \pi}{s}, \pi\right)\right)}{s}}\right)}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (*
      u
      (+
       (/ 1.0 (+ 1.0 (exp (/ PI (- s)))))
       (/
        -1.0
        (+
         2.0
         (/
          (fma
           0.16666666666666666
           (/ (* PI (* PI PI)) (* s s))
           (fma 0.5 (/ (* PI PI) s) PI))
          s))))))))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + (1.0f / (u * ((1.0f / (1.0f + expf((((float) M_PI) / -s)))) + (-1.0f / (2.0f + (fmaf(0.16666666666666666f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) / (s * s)), fmaf(0.5f, ((((float) M_PI) * ((float) M_PI)) / s), ((float) M_PI))) / s))))))));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(-1.0) / Float32(Float32(2.0) + Float32(fma(Float32(0.16666666666666666), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) / Float32(s * s)), fma(Float32(0.5), Float32(Float32(Float32(pi) * Float32(pi)) / s), Float32(pi))) / s)))))))))
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

5. Simplified 99.1%

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

6. Taylor expanded in s around -inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}, u, u\right)}\right)} - 1\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)\right)}\right)}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}, u, u\right)}\right)} - 1\right) \]

   2. unsub-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}, u, u\right)}\right)} - 1\right) \]

   3. lower--.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}\right)}}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}, u, u\right)}\right)} - 1\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{\mathsf{neg}\left(s\right)}}} + \frac{-1}{1 + \left(1 - \color{blue}{\frac{-1 \cdot \mathsf{PI}\left(\right) + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{s} + \frac{1}{2} \cdot {\mathsf{PI}\left(\right)}^{2}}{s}}{s}}\right)}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\mathsf{PI}\left(\right)}{s}}, u, u\right)}\right)} - 1\right) \]

8. Simplified 98.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{1 + \color{blue}{\left(1 - \frac{\left(-\pi\right) - \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s}, 0.5 \cdot \left(\pi \cdot \pi\right)\right)}{s}}{s}\right)}}\right) + \frac{1}{\mathsf{fma}\left(e^{\frac{\pi}{s}}, u, u\right)}\right)} - 1\right) \]

9. Taylor expanded in u around inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{-1 \cdot \frac{\mathsf{PI}\left(\right)}{s}}} - \frac{1}{2 - -1 \cdot \frac{\mathsf{PI}\left(\right) + \left(\frac{1}{6} \cdot \frac{{\mathsf{PI}\left(\right)}^{3}}{{s}^{2}} + \frac{1}{2} \cdot \frac{{\mathsf{PI}\left(\right)}^{2}}{s}\right)}{s}}\right)}} - 1\right) \]

11. Simplified 97.1%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{2 + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s \cdot s}, \mathsf{fma}\left(0.5, \frac{\pi \cdot \pi}{s}, \pi\right)\right)}{s}}\right)}} - 1\right) \]

12. Final simplification 97.1%

    \[\leadsto \left(-s\right) \cdot \log \left(-1 + \frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{-s}}} + \frac{-1}{2 + \frac{\mathsf{fma}\left(0.16666666666666666, \frac{\pi \cdot \left(\pi \cdot \pi\right)}{s \cdot s}, \mathsf{fma}\left(0.5, \frac{\pi \cdot \pi}{s}, \pi\right)\right)}{s}}\right)}\right) \]
13. Add Preprocessing

Alternative 6: 24.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(-1 + \mathsf{fma}\left(4, \frac{\pi \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)}{s}, 2\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (+ -1.0 (fma 4.0 (/ (* PI (fma -0.5 u 0.25)) s) 2.0)))))

C

float code(float u, float s) {
	return -s * logf((-1.0f + fmaf(4.0f, ((((float) M_PI) * fmaf(-0.5f, u, 0.25f)) / s), 2.0f)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + fma(Float32(4.0), Float32(Float32(Float32(pi) * fma(Float32(-0.5), u, Float32(0.25))) / s), Float32(2.0)))))
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right)}}{s} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 4.0

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

5. Simplified 4.0%

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

6. Taylor expanded in s around -inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\color{blue}{\left(2 + 4 \cdot \frac{\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} - 1\right) \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\left(2 + \color{blue}{\frac{4 \cdot \left(\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}}\right) - 1\right) \]

   2. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\color{blue}{\left(\frac{4 \cdot \left(\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s} + 2\right)} - 1\right) \]

   3. associate-*r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\left(\color{blue}{4 \cdot \frac{\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}} + 2\right) - 1\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\color{blue}{\mathsf{fma}\left(4, \frac{\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}, 2\right)} - 1\right) \]

   5. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\mathsf{fma}\left(4, \color{blue}{\frac{\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)}{s}}, 2\right) - 1\right) \]

   6. cancel-sign-sub-inv N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\color{blue}{\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)}}{s}, 2\right) - 1\right) \]

   7. associate-*r* N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\color{blue}{\left(\frac{-1}{2} \cdot u\right) \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)}{s}, 2\right) - 1\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\left(\frac{-1}{2} \cdot u\right) \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{1}{4}} \cdot \mathsf{PI}\left(\right)}{s}, 2\right) - 1\right) \]

   9. distribute-rgt-out N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)}}{s}, 2\right) - 1\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)}}{s}, 2\right) - 1\right) \]

   11. lower-PI.f32 N/A

       \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{-1}{2} \cdot u + \frac{1}{4}\right)}{s}, 2\right) - 1\right) \]

   12. lower-fma.f32 25.1

       \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \color{blue}{\mathsf{fma}\left(-0.5, u, 0.25\right)}}{s}, 2\right) - 1\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\mathsf{fma}\left(4, \frac{\pi \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)}{s}, 2\right)} - 1\right) \]

9. Final simplification 25.1%

   \[\leadsto \left(-s\right) \cdot \log \left(-1 + \mathsf{fma}\left(4, \frac{\pi \cdot \mathsf{fma}\left(-0.5, u, 0.25\right)}{s}, 2\right)\right) \]
10. Add Preprocessing

Alternative 7: 11.6% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right)}}{s} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 4.0

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

5. Simplified 4.0%

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

6. Taylor expanded in s around inf

   \[\leadsto \color{blue}{4 \cdot \left(\frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{4 \cdot \left(\frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto 4 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot u\right) \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto 4 \cdot \left(\left(\frac{1}{2} \cdot u\right) \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{-1}{4}} \cdot \mathsf{PI}\left(\right)\right) \]

   5. distribute-rgt-out N/A

      \[\leadsto 4 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto 4 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right)} \]

   7. lower-PI.f32 N/A

      \[\leadsto 4 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right) \]

   8. lower-fma.f32 11.8

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

8. Simplified 11.8%

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

9. Taylor expanded in u around inf

   \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(u \cdot \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{u}\right)\right)}\right) \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(u \cdot \left(\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{u}\right)\right)}\right) \]

    2. sub-neg N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(u \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{u}\right)\right)\right)}\right)\right) \]

    3. lower-+.f32 N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(u \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{u}\right)\right)\right)}\right)\right) \]

    4. associate-*r/ N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(u \cdot \left(\frac{1}{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{u}}\right)\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(u \cdot \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{4}}}{u}\right)\right)\right)\right)\right) \]

    6. distribute-neg-frac N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(u \cdot \left(\frac{1}{2} + \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{4}\right)}{u}}\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(u \cdot \left(\frac{1}{2} + \frac{\color{blue}{\frac{-1}{4}}}{u}\right)\right)\right) \]

    8. lower-/.f32 11.9

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

11. Simplified 11.9%

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

Alternative 8: 11.6% accurate, 42.5× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \mathsf{fma}\left(2, u, -1\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* PI (fma 2.0 u -1.0)))

C

float code(float u, float s) {
	return ((float) M_PI) * fmaf(2.0f, u, -1.0f);
}

Julia

function code(u, s)
	return Float32(Float32(pi) * fma(Float32(2.0), u, Float32(-1.0)))
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right)}}{s} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 4.0

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

5. Simplified 4.0%

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

6. Taylor expanded in s around inf

   \[\leadsto \color{blue}{4 \cdot \left(\frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{4 \cdot \left(\frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto 4 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto 4 \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot u\right) \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \mathsf{PI}\left(\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto 4 \cdot \left(\left(\frac{1}{2} \cdot u\right) \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{-1}{4}} \cdot \mathsf{PI}\left(\right)\right) \]

   5. distribute-rgt-out N/A

      \[\leadsto 4 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto 4 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right)} \]

   7. lower-PI.f32 N/A

      \[\leadsto 4 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{2} \cdot u + \frac{-1}{4}\right)\right) \]

   8. lower-fma.f32 11.8

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

8. Simplified 11.8%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. distribute-rgt-out N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 N/A

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

    6. lower-fma.f32 11.8

       \[\leadsto \pi \cdot \color{blue}{\mathsf{fma}\left(2, u, -1\right)} \]

11. Simplified 11.8%

    \[\leadsto \color{blue}{\pi \cdot \mathsf{fma}\left(2, u, -1\right)} \]
12. Add Preprocessing

Alternative 9: 11.4% accurate, 170.0× 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 99.1%

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

3. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   2. lower-neg.f32 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \]

   3. lower-PI.f32 11.7

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

5. Simplified 11.7%

   \[\leadsto \color{blue}{-\pi} \]
6. Add Preprocessing

Alternative 10: 10.4% accurate, 510.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}{s}} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \left(\frac{1}{\frac{u \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{4} - \frac{-1}{4}\right)\right)}}{s} + \frac{1}{1 + e^{\frac{\mathsf{PI}\left(\right)}{s}}}} - 1\right) \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 4.0

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

5. Simplified 4.0%

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

6. Taylor expanded in s around inf

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

8. Simplified 10.0%

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

   1. lift-neg.f32 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \color{blue}{0} \]

   4. mul0-rgt 10.0

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 10.0%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))