Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 6.8s

Alternatives: 14

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: 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{\frac{\pi}{s}} + 1}\\ \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(u, e^{-\mathsf{log1p}\left(e^{\frac{-\pi}{s}}\right)} - t\_0, t\_0\right)} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ PI s)) 1.0))))
   (*
    (- s)
    (log
     (-
      (/ 1.0 (fma u (- (exp (- (log1p (exp (/ (- PI) s))))) t_0) t_0))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (expf((((float) M_PI) / s)) + 1.0f);
	return -s * logf(((1.0f / fmaf(u, (expf(-log1pf(expf((-((float) M_PI) / s)))) - t_0), t_0)) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(exp(Float32(Float32(pi) / s)) + Float32(1.0)))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / fma(u, Float32(exp(Float32(-log1p(exp(Float32(Float32(-Float32(pi)) / s))))) - t_0), t_0)) - Float32(1.0))))
end

Derivation

1. Initial program 99.0%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. inv-pow N/A

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

   8. pow-to-exp N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-log1p.f32 N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. lift-exp.f32 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around 0

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

   1. mul-1-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-frac-neg N/A

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

   4. lower-neg.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. lift-log1p.f32 99.0

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

7. Applied rewrites 99.0%

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

9. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

Alternative 3: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. mul-1-neg N/A

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

   5. distribute-frac-neg N/A

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

   6. lower--.f32 N/A

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

5. Applied rewrites 98.0%

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

Alternative 4: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.0%

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

Alternative 5: 96.7% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     (/ 1.0 u)
     (-
      (/ 1.0 (+ (exp (/ (- PI) s)) 1.0))
      (/ 1.0 (+ 2.0 (fma 0.5 (/ (* PI PI) (* s s)) (/ 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)) - (1.0f / (2.0f + fmaf(0.5f, ((((float) M_PI) * ((float) M_PI)) / (s * s)), (((float) M_PI) / s)))))) - 1.0f));
}

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. mul-1-neg N/A

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

   5. distribute-frac-neg N/A

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

   6. lower--.f32 N/A

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-PI.f32 97.4

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

8. Applied rewrites 97.4%

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

Alternative 6: 94.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     (/ 1.0 u)
     (- (/ 1.0 (+ (exp (/ (- PI) s)) 1.0)) (/ 1.0 (+ 2.0 (/ 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)) - (1.0f / (2.0f + (((float) M_PI) / s))))) - 1.0f));
}

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in u around inf

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. mul-1-neg N/A

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

   5. distribute-frac-neg N/A

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

   6. lower--.f32 N/A

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

5. Applied rewrites 98.0%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 96.0

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

8. Applied rewrites 96.0%

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

Alternative 7: 37.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 38.0%

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

8. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-PI.f32 38.0

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

10. Applied rewrites 38.0%

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

Alternative 8: 36.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi \cdot \pi}{s}\\ \left(-s\right) \cdot \log \left(\frac{1}{\mathsf{fma}\left(0.5 - \frac{1}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.5, t\_0, -\pi\right)}{s}, 1\right) + 1}, u, \frac{1}{2 - \frac{\mathsf{fma}\left(-1, \pi, -0.5 \cdot t\_0\right)}{s}}\right)} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ (* PI PI) s)))
   (*
    (- s)
    (log
     (-
      (/
       1.0
       (fma
        (- 0.5 (/ 1.0 (+ (fma -1.0 (/ (fma -0.5 t_0 (- PI)) s) 1.0) 1.0)))
        u
        (/ 1.0 (- 2.0 (/ (fma -1.0 PI (* -0.5 t_0)) s)))))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = (((float) M_PI) * ((float) M_PI)) / s;
	return -s * logf(((1.0f / fmaf((0.5f - (1.0f / (fmaf(-1.0f, (fmaf(-0.5f, t_0, -((float) M_PI)) / s), 1.0f) + 1.0f))), u, (1.0f / (2.0f - (fmaf(-1.0f, ((float) M_PI), (-0.5f * t_0)) / s))))) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(Float32(pi) * Float32(pi)) / s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / fma(Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(fma(Float32(-1.0), Float32(fma(Float32(-0.5), t_0, Float32(-Float32(pi))) / s), Float32(1.0)) + Float32(1.0)))), u, Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(fma(Float32(-1.0), Float32(pi), Float32(Float32(-0.5) * t_0)) / s))))) - Float32(1.0))))
end

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 38.0%

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

8. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. mul-1-neg N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-PI.f32 38.0

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

10. Applied rewrites 38.0%

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

11. Taylor expanded in s around -inf

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

    1. lower-+.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-fma.f32 N/A

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

    5. lift-PI.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. pow2 N/A

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

    8. lift-*.f32 N/A

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

    9. lift-PI.f32 N/A

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

    10. lift-PI.f32 N/A

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

    11. lift-/.f32 37.7

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

13. Applied rewrites 37.7%

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

14. Final simplification 37.7%

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

Alternative 9: 36.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 38.0%

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

8. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. pow2 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. mul-1-neg N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-PI.f32 38.0

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

10. Applied rewrites 38.0%

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

11. Taylor expanded in s around inf

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

    1. lower-+.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. lift-PI.f32 37.1

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

13. Applied rewrites 37.1%

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

Alternative 10: 25.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 24.9%

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

6. Taylor expanded in u around 0

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

   1. lower-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-PI.f32 25.1

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

8. Applied rewrites 25.1%

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

Alternative 11: 11.6% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

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

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

   6. metadata-eval N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

      \[\leadsto 4 \cdot \mathsf{fma}\left(u, \pi \cdot \frac{1}{2}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   9. lift-*.f32 N/A

      \[\leadsto 4 \cdot \mathsf{fma}\left(u, \pi \cdot \frac{1}{2}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   10. lift-PI.f32 10.8

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

7. Applied rewrites 10.8%

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

8. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

      \[\leadsto u \cdot \mathsf{fma}\left(-1, \frac{\mathsf{PI}\left(\right)}{u}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

   4. lift-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 10.8

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

10. Applied rewrites 10.8%

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

11. Final simplification 10.8%

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

Alternative 12: 11.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 10.8%

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

Alternative 13: 11.6% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

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

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

   6. metadata-eval N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

      \[\leadsto 4 \cdot \mathsf{fma}\left(u, \pi \cdot \frac{1}{2}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   9. lift-*.f32 N/A

      \[\leadsto 4 \cdot \mathsf{fma}\left(u, \pi \cdot \frac{1}{2}, \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) \]

   10. lift-PI.f32 10.8

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

7. Applied rewrites 10.8%

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

8. Taylor expanded in u around 0

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

   1. lower-fma.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 10.8

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

10. Applied rewrites 10.8%

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

11. Final simplification 10.8%

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

Alternative 14: 11.3% 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 99.0%

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

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

   2. lift-neg.f32 N/A

      \[\leadsto -\mathsf{PI}\left(\right) \]

   3. lift-PI.f32 10.7

      \[\leadsto -\pi \]

5. Applied rewrites 10.7%

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

Reproduce

herbie shell --seed 2025085 
(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))))