Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 5.9s

Alternatives: 9

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	return Float32(Float32(-s) * log(Float32((fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - Float32(Float32(1.0) / Float32(Float32(1.0) + t_0))), exp(Float32(-log1p(t_0)))) ^ 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. inv-pow N/A

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

   7. pow-to-exp N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-exp.f32 99.0

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in s around 0

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

   1. log-pow-rev N/A

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

   2. inv-pow N/A

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

   3. log-rec N/A

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

   4. lower-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-log1p.f32 99.0

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 99.0%

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

Alternative 2: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(u, \frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + t\_0}, e^{-\mathsf{log1p}\left(t\_0\right)}\right)\right) \cdot -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s))))
   (*
    (- s)
    (log
     (expm1
      (*
       (log
        (fma
         u
         (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 t_0)))
         (exp (- (log1p t_0)))))
       -1.0))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	return -s * logf(expm1f((logf(fmaf(u, ((1.0f / (1.0f + expf((-((float) M_PI) / s)))) - (1.0f / (1.0f + t_0))), expf(-log1pf(t_0)))) * -1.0f)));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	return Float32(Float32(-s) * log(expm1(Float32(log(fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - Float32(Float32(1.0) / Float32(Float32(1.0) + t_0))), exp(Float32(-log1p(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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. inv-pow N/A

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

   7. pow-to-exp N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-exp.f32 99.0

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in s around 0

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

   1. log-pow-rev N/A

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

   2. inv-pow N/A

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

   3. log-rec N/A

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

   4. lower-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-log1p.f32 99.0

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 99.0%

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

   1. lift--.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. pow-to-exp N/A

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

   4. lower-expm1.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(u, \frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, e^{-\mathsf{log1p}\left(e^{\frac{\pi}{s}}\right)}\right)\right) \cdot -1\right)\right)} \]

10. Applied rewrites 98.9%

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(u, \frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, e^{-\mathsf{log1p}\left(e^{\frac{\pi}{s}}\right)}\right)\right) \cdot -1\right)\right)} \]
11. Add Preprocessing

Alternative 3: 97.6% 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. Taylor expanded in u around inf

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

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

4. Applied rewrites 97.6%

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

Alternative 4: 37.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ \left(-s\right) \cdot \log \left({\left(\mathsf{fma}\left(u, 0.5 - \frac{1}{1 + t\_0}, e^{-\mathsf{log1p}\left(t\_0\right)}\right)\right)}^{-1} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s))))
   (*
    (- s)
    (log
     (-
      (pow (fma u (- 0.5 (/ 1.0 (+ 1.0 t_0))) (exp (- (log1p t_0)))) -1.0)
      1.0)))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	return -s * logf((powf(fmaf(u, (0.5f - (1.0f / (1.0f + t_0))), expf(-log1pf(t_0))), -1.0f) - 1.0f));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	return Float32(Float32(-s) * log(Float32((fma(u, Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(Float32(1.0) + t_0))), exp(Float32(-log1p(t_0)))) ^ 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. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. inv-pow N/A

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

   7. pow-to-exp N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-log1p.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-exp.f32 99.0

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in s around 0

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

   1. log-pow-rev N/A

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

   2. inv-pow N/A

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

   3. log-rec N/A

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

   4. lower-neg.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. lift-log1p.f32 99.0

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left({\left(\mathsf{fma}\left(u, \color{blue}{\frac{1}{2}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, e^{-\mathsf{log1p}\left(e^{\frac{\pi}{s}}\right)}\right)\right)}^{-1} - 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 37.8%

    \[\leadsto \left(-s\right) \cdot \log \left({\left(\mathsf{fma}\left(u, \color{blue}{0.5} - \frac{1}{1 + e^{\frac{\pi}{s}}}, e^{-\mathsf{log1p}\left(e^{\frac{\pi}{s}}\right)}\right)\right)}^{-1} - 1\right) \]
12. Add Preprocessing

Alternative 5: 37.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(0.5 - \frac{1}{2 + \frac{\pi}{s}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (expm1
    (*
     (log
      (fma (- 0.5 (/ 1.0 (+ 2.0 (/ PI s)))) u (/ 1.0 (+ (exp (/ PI s)) 1.0))))
     -1.0)))))

C

float code(float u, float s) {
	return -s * logf(expm1f((logf(fmaf((0.5f - (1.0f / (2.0f + (((float) M_PI) / s)))), u, (1.0f / (expf((((float) M_PI) / s)) + 1.0f)))) * -1.0f)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(expm1(Float32(log(fma(Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(pi) / s)))), u, Float32(Float32(1.0) / Float32(exp(Float32(Float32(pi) / s)) + Float32(1.0))))) * Float32(-1.0)))))
end

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 98.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{1}{e^{\frac{-\pi}{s}} + 1} - \frac{1}{e^{\frac{\pi}{s}} + 1}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right)} \]

4. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\color{blue}{\frac{1}{2}} - \frac{1}{e^{\frac{\pi}{s}} + 1}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 37.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\color{blue}{0.5} - \frac{1}{e^{\frac{\pi}{s}} + 1}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

7. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]
8. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

   2. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{\color{blue}{s}}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

   3. lift-PI.f32 37.8

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(0.5 - \frac{1}{2 + \frac{\pi}{s}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

9. Applied rewrites 37.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(0.5 - \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]
10. Add Preprocessing

Alternative 6: 37.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(0.5 - \frac{1}{\frac{\pi}{s}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (expm1
    (*
     (log (fma (- 0.5 (/ 1.0 (/ PI s))) u (/ 1.0 (+ (exp (/ PI s)) 1.0))))
     -1.0)))))

C

float code(float u, float s) {
	return -s * logf(expm1f((logf(fmaf((0.5f - (1.0f / (((float) M_PI) / s))), u, (1.0f / (expf((((float) M_PI) / s)) + 1.0f)))) * -1.0f)));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(expm1(Float32(log(fma(Float32(Float32(0.5) - Float32(Float32(1.0) / Float32(Float32(pi) / s))), u, Float32(Float32(1.0) / Float32(exp(Float32(Float32(pi) / s)) + Float32(1.0))))) * Float32(-1.0)))))
end

Derivation

1. Initial program 99.0%

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

3. Applied rewrites 98.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{1}{e^{\frac{-\pi}{s}} + 1} - \frac{1}{e^{\frac{\pi}{s}} + 1}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right)} \]

4. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\color{blue}{\frac{1}{2}} - \frac{1}{e^{\frac{\pi}{s}} + 1}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 37.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\color{blue}{0.5} - \frac{1}{e^{\frac{\pi}{s}} + 1}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

7. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{\color{blue}{2 + \frac{\mathsf{PI}\left(\right)}{s}}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]
8. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

   2. lift-/.f32 N/A

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2 + \frac{\mathsf{PI}\left(\right)}{\color{blue}{s}}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

   3. lift-PI.f32 37.8

      \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(0.5 - \frac{1}{2 + \frac{\pi}{s}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

9. Applied rewrites 37.8%

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(0.5 - \frac{1}{\color{blue}{2 + \frac{\pi}{s}}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

10. Taylor expanded in s around 0

    \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{\frac{\mathsf{PI}\left(\right)}{\color{blue}{s}}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]
11. Step-by-step derivation

    1. lift-/.f32 N/A

       \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{\frac{\mathsf{PI}\left(\right)}{s}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

    2. lift-PI.f32 37.8

       \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(0.5 - \frac{1}{\frac{\pi}{s}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]

12. Applied rewrites 37.8%

    \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(0.5 - \frac{1}{\frac{\pi}{\color{blue}{s}}}, u, \frac{1}{e^{\frac{\pi}{s}} + 1}\right)\right) \cdot -1\right)\right) \]
13. Add Preprocessing

Alternative 7: 25.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((s + 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. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
3. 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

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

4. Applied rewrites 25.0%

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

5. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 25.0

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

7. Applied rewrites 25.0%

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

8. Taylor expanded in u around 0

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

   1. lift-PI.f32 25.2

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

10. Applied rewrites 25.2%

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

Alternative 8: 25.2% accurate, 4.1× 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. Taylor expanded in s around inf

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
3. 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

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

4. Applied rewrites 25.0%

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

5. Taylor expanded in u around 0

   \[\leadsto \left(-s\right) \cdot \log \left(1 + \color{blue}{\frac{\mathsf{PI}\left(\right)}{s}}\right) \]
6. 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 25.2

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

7. Applied rewrites 25.2%

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

Alternative 9: 11.4% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{-1 \cdot \mathsf{PI}\left(\right)} \]
3. 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 11.4

      \[\leadsto -\pi \]

4. Applied rewrites 11.4%

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

Reproduce

herbie shell --seed 2025101 
(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))))