Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 16.5s

Alternatives: 9

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

5. Applied rewrites 98.8%

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

6. Final simplification 98.8%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

4. Applied rewrites 98.7%

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

6. Applied rewrites 98.8%

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

7. Final simplification 98.8%

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

Alternative 3: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in s around 0

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-exp.f32 N/A

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

   10. mul-1-neg N/A

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

   11. distribute-neg-frac2 N/A

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

   12. mul-1-neg N/A

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

   13. lower-/.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f32 98.7

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

10. Applied rewrites 98.7%

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

11. Final simplification 98.7%

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

Alternative 4: 97.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in u around inf

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

10. Applied rewrites 96.6%

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

11. Final simplification 96.6%

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

Alternative 5: 11.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(pi) * log(Float32(exp(1)))) * fma(Float32(0.5), u, Float32(-0.25))))
end

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 3.3

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

5. Applied rewrites 3.3%

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

6. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

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

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-fma.f32 11.7

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

8. Applied rewrites 11.7%

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

   1. add-log-exp N/A

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

   2. *-un-lft-identity N/A

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

   3. lift-PI.f32 N/A

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

   4. exp-prod N/A

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

   5. log-pow N/A

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

   6. lower-*.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. exp-1-e N/A

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

   9. lower-E.f32 11.7

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

10. Applied rewrites 11.7%

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

Alternative 6: 11.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.25, u \cdot u, 0.0625\right)\\ 4 \cdot \left(\pi \cdot \frac{\mathsf{fma}\left(u \cdot \left(u \cdot u\right), 0.125, -0.015625\right)}{\frac{t\_0 \cdot t\_0 - \left(u \cdot 0.125\right) \cdot \left(u \cdot 0.125\right)}{t\_0 - u \cdot 0.125}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (fma 0.25 (* u u) 0.0625)))
   (*
    4.0
    (*
     PI
     (/
      (fma (* u (* u u)) 0.125 -0.015625)
      (/ (- (* t_0 t_0) (* (* u 0.125) (* u 0.125))) (- t_0 (* u 0.125))))))))

C

float code(float u, float s) {
	float t_0 = fmaf(0.25f, (u * u), 0.0625f);
	return 4.0f * (((float) M_PI) * (fmaf((u * (u * u)), 0.125f, -0.015625f) / (((t_0 * t_0) - ((u * 0.125f) * (u * 0.125f))) / (t_0 - (u * 0.125f)))));
}

Julia

function code(u, s)
	t_0 = fma(Float32(0.25), Float32(u * u), Float32(0.0625))
	return Float32(Float32(4.0) * Float32(Float32(pi) * Float32(fma(Float32(u * Float32(u * u)), Float32(0.125), Float32(-0.015625)) / Float32(Float32(Float32(t_0 * t_0) - Float32(Float32(u * Float32(0.125)) * Float32(u * Float32(0.125)))) / Float32(t_0 - Float32(u * Float32(0.125)))))))
end

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 3.3

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

5. Applied rewrites 3.3%

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

6. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

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

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-fma.f32 11.7

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

8. Applied rewrites 11.7%

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

   1. flip3-+ N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{{\left(\frac{1}{2} \cdot u\right)}^{3} + {\frac{-1}{4}}^{3}}{\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \left(\frac{-1}{4} \cdot \frac{-1}{4} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}\right)}}\right) \]

   2. lower-/.f32 N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{{\left(\frac{1}{2} \cdot u\right)}^{3} + {\frac{-1}{4}}^{3}}{\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \left(\frac{-1}{4} \cdot \frac{-1}{4} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}\right)}}\right) \]

   3. *-commutative N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{{\color{blue}{\left(u \cdot \frac{1}{2}\right)}}^{3} + {\frac{-1}{4}}^{3}}{\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \left(\frac{-1}{4} \cdot \frac{-1}{4} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}\right)}\right) \]

   4. unpow-prod-down N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\color{blue}{{u}^{3} \cdot {\frac{1}{2}}^{3}} + {\frac{-1}{4}}^{3}}{\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \left(\frac{-1}{4} \cdot \frac{-1}{4} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}\right)}\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\color{blue}{\mathsf{fma}\left({u}^{3}, {\frac{1}{2}}^{3}, {\frac{-1}{4}}^{3}\right)}}{\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \left(\frac{-1}{4} \cdot \frac{-1}{4} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}\right)}\right) \]

   6. unpow3 N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(u \cdot u\right) \cdot u}, {\frac{1}{2}}^{3}, {\frac{-1}{4}}^{3}\right)}{\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \left(\frac{-1}{4} \cdot \frac{-1}{4} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}\right)}\right) \]

   7. lower-*.f32 N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(u \cdot u\right) \cdot u}, {\frac{1}{2}}^{3}, {\frac{-1}{4}}^{3}\right)}{\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \left(\frac{-1}{4} \cdot \frac{-1}{4} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}\right)}\right) \]

   8. lower-*.f32 N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(u \cdot u\right)} \cdot u, {\frac{1}{2}}^{3}, {\frac{-1}{4}}^{3}\right)}{\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \left(\frac{-1}{4} \cdot \frac{-1}{4} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \color{blue}{\frac{1}{8}}, {\frac{-1}{4}}^{3}\right)}{\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \left(\frac{-1}{4} \cdot \frac{-1}{4} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}\right)}\right) \]

   10. metadata-eval N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \color{blue}{\frac{-1}{64}}\right)}{\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \left(\frac{-1}{4} \cdot \frac{-1}{4} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}\right)}\right) \]

   11. associate-+r- N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\color{blue}{\left(\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \frac{-1}{4} \cdot \frac{-1}{4}\right) - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}}}\right) \]

   12. lower--.f32 N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\color{blue}{\left(\left(\frac{1}{2} \cdot u\right) \cdot \left(\frac{1}{2} \cdot u\right) + \frac{-1}{4} \cdot \frac{-1}{4}\right) - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}}}\right) \]

   13. swap-sqr N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \left(u \cdot u\right)} + \frac{-1}{4} \cdot \frac{-1}{4}\right) - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}}\right) \]

   14. metadata-eval N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\left(\color{blue}{\frac{1}{4}} \cdot \left(u \cdot u\right) + \frac{-1}{4} \cdot \frac{-1}{4}\right) - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}}\right) \]

   15. metadata-eval N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot \left(u \cdot u\right) + \frac{-1}{4} \cdot \frac{-1}{4}\right) - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}}\right) \]

   16. lower-fma.f32 N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{-1}{4}\right), u \cdot u, \frac{-1}{4} \cdot \frac{-1}{4}\right)} - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}}\right) \]

   17. metadata-eval N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\mathsf{fma}\left(\color{blue}{\frac{1}{4}}, u \cdot u, \frac{-1}{4} \cdot \frac{-1}{4}\right) - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}}\right) \]

   18. lower-*.f32 N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{u \cdot u}, \frac{-1}{4} \cdot \frac{-1}{4}\right) - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}}\right) \]

   19. metadata-eval N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \color{blue}{\frac{1}{16}}\right) - \left(\frac{1}{2} \cdot u\right) \cdot \frac{-1}{4}}\right) \]

   20. *-commutative N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) - \color{blue}{\left(u \cdot \frac{1}{2}\right)} \cdot \frac{-1}{4}}\right) \]

   21. associate-*l* N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) - \color{blue}{u \cdot \left(\frac{1}{2} \cdot \frac{-1}{4}\right)}}\right) \]

   22. lower-*.f32 N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) - \color{blue}{u \cdot \left(\frac{1}{2} \cdot \frac{-1}{4}\right)}}\right) \]

   23. metadata-eval 11.7

       \[\leadsto 4 \cdot \left(\pi \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, 0.125, -0.015625\right)}{\mathsf{fma}\left(0.25, u \cdot u, 0.0625\right) - u \cdot \color{blue}{-0.125}}\right) \]

10. Applied rewrites 11.7%

    \[\leadsto 4 \cdot \left(\pi \cdot \color{blue}{\frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, 0.125, -0.015625\right)}{\mathsf{fma}\left(0.25, u \cdot u, 0.0625\right) - u \cdot -0.125}}\right) \]
11. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\left(\frac{1}{4} \cdot \color{blue}{\left(u \cdot u\right)} + \frac{1}{16}\right) - u \cdot \frac{-1}{8}}\right) \]

    2. lift-fma.f32 N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right)} - u \cdot \frac{-1}{8}}\right) \]

    3. lift-*.f32 N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) - \color{blue}{u \cdot \frac{-1}{8}}}\right) \]

    4. sub-neg N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) + \left(\mathsf{neg}\left(u \cdot \frac{-1}{8}\right)\right)}}\right) \]

    5. flip-+ N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) \cdot \mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) - \left(\mathsf{neg}\left(u \cdot \frac{-1}{8}\right)\right) \cdot \left(\mathsf{neg}\left(u \cdot \frac{-1}{8}\right)\right)}{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) - \left(\mathsf{neg}\left(u \cdot \frac{-1}{8}\right)\right)}}}\right) \]

    6. lower-/.f32 N/A

       \[\leadsto 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, \frac{1}{8}, \frac{-1}{64}\right)}{\color{blue}{\frac{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) \cdot \mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) - \left(\mathsf{neg}\left(u \cdot \frac{-1}{8}\right)\right) \cdot \left(\mathsf{neg}\left(u \cdot \frac{-1}{8}\right)\right)}{\mathsf{fma}\left(\frac{1}{4}, u \cdot u, \frac{1}{16}\right) - \left(\mathsf{neg}\left(u \cdot \frac{-1}{8}\right)\right)}}}\right) \]

12. Applied rewrites 11.7%

    \[\leadsto 4 \cdot \left(\pi \cdot \frac{\mathsf{fma}\left(\left(u \cdot u\right) \cdot u, 0.125, -0.015625\right)}{\color{blue}{\frac{\mathsf{fma}\left(0.25, u \cdot u, 0.0625\right) \cdot \mathsf{fma}\left(0.25, u \cdot u, 0.0625\right) - \left(u \cdot 0.125\right) \cdot \left(u \cdot 0.125\right)}{\mathsf{fma}\left(0.25, u \cdot u, 0.0625\right) - u \cdot 0.125}}}\right) \]

13. Final simplification 11.7%

    \[\leadsto 4 \cdot \left(\pi \cdot \frac{\mathsf{fma}\left(u \cdot \left(u \cdot u\right), 0.125, -0.015625\right)}{\frac{\mathsf{fma}\left(0.25, u \cdot u, 0.0625\right) \cdot \mathsf{fma}\left(0.25, u \cdot u, 0.0625\right) - \left(u \cdot 0.125\right) \cdot \left(u \cdot 0.125\right)}{\mathsf{fma}\left(0.25, u \cdot u, 0.0625\right) - u \cdot 0.125}}\right) \]
14. Add Preprocessing

Alternative 7: 11.6% accurate, 42.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 3.3

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

5. Applied rewrites 3.3%

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

6. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

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

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-fma.f32 11.7

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

8. Applied rewrites 11.7%

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

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. distribute-rgt-out N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 N/A

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

    6. lower-fma.f32 11.7

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

11. Applied rewrites 11.7%

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

Alternative 8: 11.4% accurate, 170.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in u around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-PI.f32 11.5

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

5. Applied rewrites 11.5%

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

Alternative 9: 4.6% accurate, 510.0× 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(pi)
end

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 3.3

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

5. Applied rewrites 3.3%

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

7. Applied rewrites 0.6%

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

8. Taylor expanded in u around 0

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

   1. lower-PI.f32 4.7

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

10. Applied rewrites 4.7%

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

Reproduce

herbie shell --seed 2024220 
(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))))