Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 59.2s

Alternatives: 7

Speedup: N/A×

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, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}} + 1\\ t_1 := \frac{1}{t\_0}\\ t_2 := \left(\frac{1}{{\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)} + 1} - t\_1\right) \cdot u\\ t_3 := t\_2 \cdot t\_2\\ t_4 := \frac{-1}{t\_0}\\ \left(-1 \cdot s\right) \cdot \log \left(\frac{1}{\frac{\mathsf{fma}\left(t\_2, t\_3, \frac{1}{{t\_0}^{3}}\right)}{t\_3 + \left(t\_4 \cdot t\_4 - t\_2 \cdot t\_1\right)}} - 1\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ (exp (/ PI s)) 1.0))
        (t_1 (/ 1.0 t_0))
        (t_2 (* (- (/ 1.0 (+ (pow (exp -1.0) (/ PI s)) 1.0)) t_1) u))
        (t_3 (* t_2 t_2))
        (t_4 (/ -1.0 t_0)))
   (*
    (* -1.0 s)
    (log
     (-
      (/
       1.0
       (/
        (fma t_2 t_3 (/ 1.0 (pow t_0 3.0)))
        (+ t_3 (- (* t_4 t_4) (* t_2 t_1)))))
      1.0)))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s)) + 1.0f;
	float t_1 = 1.0f / t_0;
	float t_2 = ((1.0f / (powf(expf(-1.0f), (((float) M_PI) / s)) + 1.0f)) - t_1) * u;
	float t_3 = t_2 * t_2;
	float t_4 = -1.0f / t_0;
	return (-1.0f * s) * logf(((1.0f / (fmaf(t_2, t_3, (1.0f / powf(t_0, 3.0f))) / (t_3 + ((t_4 * t_4) - (t_2 * t_1))))) - 1.0f));
}

Julia

function code(u, s)
	t_0 = Float32(exp(Float32(Float32(pi) / s)) + Float32(1.0))
	t_1 = Float32(Float32(1.0) / t_0)
	t_2 = Float32(Float32(Float32(Float32(1.0) / Float32((exp(Float32(-1.0)) ^ Float32(Float32(pi) / s)) + Float32(1.0))) - t_1) * u)
	t_3 = Float32(t_2 * t_2)
	t_4 = Float32(Float32(-1.0) / t_0)
	return Float32(Float32(Float32(-1.0) * s) * log(Float32(Float32(Float32(1.0) / Float32(fma(t_2, t_3, Float32(Float32(1.0) / (t_0 ^ Float32(3.0)))) / Float32(t_3 + Float32(Float32(t_4 * t_4) - Float32(t_2 * t_1))))) - Float32(1.0))))
end

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, N/A× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s))))))
   (*
    (* -1.0 s)
    (log
     (-
      (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (* -1.0 (/ 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 (-1.0f * s) * logf(((1.0f / ((u * ((1.0f / (1.0f + expf((-1.0f * (((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(Float32(-1.0) * s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-1.0) * 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 = (single(-1.0) * s) * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((single(-1.0) * (single(pi) / s))))) - t_0)) + t_0)) - single(1.0)));
end

Derivation

1. Initial program 98.9%

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

3. Final simplification 98.9%

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

Alternative 3: 98.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ t_1 := \mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - t\_0, t\_0\right)\\ \left(-1 \cdot s\right) \cdot \log \left(\frac{{t\_1}^{-3} - 1}{1 + \left(\frac{1}{t\_1} + {t\_1}^{-2}\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s)))))
        (t_1 (fma u (- (/ 1.0 (+ 1.0 (pow (exp -1.0) (/ PI s)))) t_0) t_0)))
   (*
    (* -1.0 s)
    (log (/ (- (pow t_1 -3.0) 1.0) (+ 1.0 (+ (/ 1.0 t_1) (pow t_1 -2.0))))))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float t_1 = fmaf(u, ((1.0f / (1.0f + powf(expf(-1.0f), (((float) M_PI) / s)))) - t_0), t_0);
	return (-1.0f * s) * logf(((powf(t_1, -3.0f) - 1.0f) / (1.0f + ((1.0f / t_1) + powf(t_1, -2.0f)))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	t_1 = fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(-1.0)) ^ Float32(Float32(pi) / s)))) - t_0), t_0)
	return Float32(Float32(Float32(-1.0) * s) * log(Float32(Float32((t_1 ^ Float32(-3.0)) - Float32(1.0)) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / t_1) + (t_1 ^ Float32(-2.0)))))))
end

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in s around 0

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

   \[\leadsto \left(-1 \cdot s\right) \cdot \log \left(\frac{{\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3} - 1}{1 + \left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + {\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-2}\right)}\right) \]
9. Add Preprocessing

Alternative 4: 98.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ t_1 := \mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - t\_0, t\_0\right)\\ \left(-1 \cdot s\right) \cdot \left(\log \left({t\_1}^{-2} - 1\right) - \mathsf{log1p}\left(\frac{1}{t\_1}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s)))))
        (t_1 (fma u (- (/ 1.0 (+ 1.0 (pow (exp -1.0) (/ PI s)))) t_0) t_0)))
   (* (* -1.0 s) (- (log (- (pow t_1 -2.0) 1.0)) (log1p (/ 1.0 t_1))))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float t_1 = fmaf(u, ((1.0f / (1.0f + powf(expf(-1.0f), (((float) M_PI) / s)))) - t_0), t_0);
	return (-1.0f * s) * (logf((powf(t_1, -2.0f) - 1.0f)) - log1pf((1.0f / t_1)));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	t_1 = fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(-1.0)) ^ Float32(Float32(pi) / s)))) - t_0), t_0)
	return Float32(Float32(Float32(-1.0) * s) * Float32(log(Float32((t_1 ^ Float32(-2.0)) - Float32(1.0))) - log1p(Float32(Float32(1.0) / t_1))))
end

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in s around 0

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

   \[\leadsto \left(-1 \cdot s\right) \cdot \left(\log \left({\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-2} - 1\right) - \mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}\right)\right) \]
9. Add Preprocessing

Alternative 5: 97.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ t_1 := \mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - t\_0, t\_0\right)\\ t_2 := \mathsf{log1p}\left(\frac{1}{t\_1} + {t\_1}^{-2}\right)\\ t_3 := \mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - t\_0, t\_0\right)\\ t_4 := \log \left(\frac{1}{{t\_3}^{3}} - 1\right)\\ t_5 := \mathsf{log1p}\left(\frac{1}{t\_3} + \frac{1}{{t\_3}^{2}}\right)\\ -1 \cdot \frac{s \cdot \left({t\_4}^{3} - \left(t\_2 \cdot t\_2\right) \cdot t\_2\right)}{\mathsf{fma}\left(t\_5, t\_4, {t\_5}^{2} + {t\_4}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s)))))
        (t_1 (fma u (- (/ 1.0 (+ 1.0 (pow (exp -1.0) (/ PI s)))) t_0) t_0))
        (t_2 (log1p (+ (/ 1.0 t_1) (pow t_1 -2.0))))
        (t_3 (fma u (- (/ 1.0 (+ 1.0 (exp (* -1.0 (/ PI s))))) t_0) t_0))
        (t_4 (log (- (/ 1.0 (pow t_3 3.0)) 1.0)))
        (t_5 (log1p (+ (/ 1.0 t_3) (/ 1.0 (pow t_3 2.0))))))
   (*
    -1.0
    (/
     (* s (- (pow t_4 3.0) (* (* t_2 t_2) t_2)))
     (fma t_5 t_4 (+ (pow t_5 2.0) (pow t_4 2.0)))))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float t_1 = fmaf(u, ((1.0f / (1.0f + powf(expf(-1.0f), (((float) M_PI) / s)))) - t_0), t_0);
	float t_2 = log1pf(((1.0f / t_1) + powf(t_1, -2.0f)));
	float t_3 = fmaf(u, ((1.0f / (1.0f + expf((-1.0f * (((float) M_PI) / s))))) - t_0), t_0);
	float t_4 = logf(((1.0f / powf(t_3, 3.0f)) - 1.0f));
	float t_5 = log1pf(((1.0f / t_3) + (1.0f / powf(t_3, 2.0f))));
	return -1.0f * ((s * (powf(t_4, 3.0f) - ((t_2 * t_2) * t_2))) / fmaf(t_5, t_4, (powf(t_5, 2.0f) + powf(t_4, 2.0f))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	t_1 = fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(-1.0)) ^ Float32(Float32(pi) / s)))) - t_0), t_0)
	t_2 = log1p(Float32(Float32(Float32(1.0) / t_1) + (t_1 ^ Float32(-2.0))))
	t_3 = fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s))))) - t_0), t_0)
	t_4 = log(Float32(Float32(Float32(1.0) / (t_3 ^ Float32(3.0))) - Float32(1.0)))
	t_5 = log1p(Float32(Float32(Float32(1.0) / t_3) + Float32(Float32(1.0) / (t_3 ^ Float32(2.0)))))
	return Float32(Float32(-1.0) * Float32(Float32(s * Float32((t_4 ^ Float32(3.0)) - Float32(Float32(t_2 * t_2) * t_2))) / fma(t_5, t_4, Float32((t_5 ^ Float32(2.0)) + (t_4 ^ Float32(2.0))))))
end

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 96.9%

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

7. Taylor expanded in s around 0

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

9. Applied rewrites 97.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{s \cdot \left({\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{3} - {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right), \log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right), {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{2} + {\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{2}\right)}} \]

11. Applied rewrites 97.0%

    \[\leadsto -1 \cdot \frac{s \cdot \left({\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{3} - \left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + {\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-2}\right) \cdot \mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + {\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-2}\right)\right) \cdot \mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + {\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-2}\right)\right)}{\mathsf{fma}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right), \log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right), {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{2} + {\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{2}\right)} \]
12. Add Preprocessing

Alternative 6: 96.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ t_1 := \mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - t\_0, t\_0\right)\\ t_2 := \mathsf{log1p}\left(\frac{1}{t\_1} + \frac{1}{{t\_1}^{2}}\right)\\ t_3 := \log \left({\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - t\_0, t\_0\right)\right)}^{-3} - 1\right)\\ t_4 := \log \left(\frac{1}{{t\_1}^{3}} - 1\right)\\ -1 \cdot \frac{s \cdot \left(\left(t\_3 \cdot t\_3\right) \cdot t\_3 - {t\_2}^{3}\right)}{\mathsf{fma}\left(t\_2, t\_4, {t\_2}^{2} + {t\_4}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s)))))
        (t_1 (fma u (- (/ 1.0 (+ 1.0 (exp (* -1.0 (/ PI s))))) t_0) t_0))
        (t_2 (log1p (+ (/ 1.0 t_1) (/ 1.0 (pow t_1 2.0)))))
        (t_3
         (log
          (-
           (pow
            (fma u (- (/ 1.0 (+ 1.0 (pow (exp -1.0) (/ PI s)))) t_0) t_0)
            -3.0)
           1.0)))
        (t_4 (log (- (/ 1.0 (pow t_1 3.0)) 1.0))))
   (*
    -1.0
    (/
     (* s (- (* (* t_3 t_3) t_3) (pow t_2 3.0)))
     (fma t_2 t_4 (+ (pow t_2 2.0) (pow t_4 2.0)))))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float t_1 = fmaf(u, ((1.0f / (1.0f + expf((-1.0f * (((float) M_PI) / s))))) - t_0), t_0);
	float t_2 = log1pf(((1.0f / t_1) + (1.0f / powf(t_1, 2.0f))));
	float t_3 = logf((powf(fmaf(u, ((1.0f / (1.0f + powf(expf(-1.0f), (((float) M_PI) / s)))) - t_0), t_0), -3.0f) - 1.0f));
	float t_4 = logf(((1.0f / powf(t_1, 3.0f)) - 1.0f));
	return -1.0f * ((s * (((t_3 * t_3) * t_3) - powf(t_2, 3.0f))) / fmaf(t_2, t_4, (powf(t_2, 2.0f) + powf(t_4, 2.0f))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	t_1 = fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s))))) - t_0), t_0)
	t_2 = log1p(Float32(Float32(Float32(1.0) / t_1) + Float32(Float32(1.0) / (t_1 ^ Float32(2.0)))))
	t_3 = log(Float32((fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(-1.0)) ^ Float32(Float32(pi) / s)))) - t_0), t_0) ^ Float32(-3.0)) - Float32(1.0)))
	t_4 = log(Float32(Float32(Float32(1.0) / (t_1 ^ Float32(3.0))) - Float32(1.0)))
	return Float32(Float32(-1.0) * Float32(Float32(s * Float32(Float32(Float32(t_3 * t_3) * t_3) - (t_2 ^ Float32(3.0)))) / fma(t_2, t_4, Float32((t_2 ^ Float32(2.0)) + (t_4 ^ Float32(2.0))))))
end

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 96.9%

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

7. Taylor expanded in s around 0

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

9. Applied rewrites 97.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{s \cdot \left({\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{3} - {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right), \log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right), {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{2} + {\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{2}\right)}} \]

11. Applied rewrites 96.8%

    \[\leadsto -1 \cdot \frac{s \cdot \left(\left(\log \left({\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3} - 1\right) \cdot \log \left({\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3} - 1\right)\right) \cdot \log \left({\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3} - 1\right) - {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right), \log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right), {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{2} + {\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{2}\right)} \]
12. Add Preprocessing

Alternative 7: 91.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ t_1 := \mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - t\_0, t\_0\right)\\ t_2 := \log \left(\frac{1}{{t\_1}^{3}} - 1\right)\\ t_3 := \mathsf{log1p}\left(\frac{1}{t\_1} + \frac{1}{{t\_1}^{2}}\right)\\ \frac{s \cdot \left(e^{\log \log \left({\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - t\_0, t\_0\right)\right)}^{-3} - 1\right) \cdot 3} + -1 \cdot {t\_3}^{3}\right)}{-1 \cdot \mathsf{fma}\left(t\_3, t\_2, {t\_3}^{2} + {t\_2}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ PI s)))))
        (t_1 (fma u (- (/ 1.0 (+ 1.0 (exp (* -1.0 (/ PI s))))) t_0) t_0))
        (t_2 (log (- (/ 1.0 (pow t_1 3.0)) 1.0)))
        (t_3 (log1p (+ (/ 1.0 t_1) (/ 1.0 (pow t_1 2.0))))))
   (/
    (*
     s
     (+
      (exp
       (*
        (log
         (log
          (-
           (pow
            (fma u (- (/ 1.0 (+ 1.0 (pow (exp -1.0) (/ PI s)))) t_0) t_0)
            -3.0)
           1.0)))
        3.0))
      (* -1.0 (pow t_3 3.0))))
    (* -1.0 (fma t_3 t_2 (+ (pow t_3 2.0) (pow t_2 2.0)))))))

C

float code(float u, float s) {
	float t_0 = 1.0f / (1.0f + expf((((float) M_PI) / s)));
	float t_1 = fmaf(u, ((1.0f / (1.0f + expf((-1.0f * (((float) M_PI) / s))))) - t_0), t_0);
	float t_2 = logf(((1.0f / powf(t_1, 3.0f)) - 1.0f));
	float t_3 = log1pf(((1.0f / t_1) + (1.0f / powf(t_1, 2.0f))));
	return (s * (expf((logf(logf((powf(fmaf(u, ((1.0f / (1.0f + powf(expf(-1.0f), (((float) M_PI) / s)))) - t_0), t_0), -3.0f) - 1.0f))) * 3.0f)) + (-1.0f * powf(t_3, 3.0f)))) / (-1.0f * fmaf(t_3, t_2, (powf(t_3, 2.0f) + powf(t_2, 2.0f))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))
	t_1 = fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-1.0) * Float32(Float32(pi) / s))))) - t_0), t_0)
	t_2 = log(Float32(Float32(Float32(1.0) / (t_1 ^ Float32(3.0))) - Float32(1.0)))
	t_3 = log1p(Float32(Float32(Float32(1.0) / t_1) + Float32(Float32(1.0) / (t_1 ^ Float32(2.0)))))
	return Float32(Float32(s * Float32(exp(Float32(log(log(Float32((fma(u, Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(-1.0)) ^ Float32(Float32(pi) / s)))) - t_0), t_0) ^ Float32(-3.0)) - Float32(1.0)))) * Float32(3.0))) + Float32(Float32(-1.0) * (t_3 ^ Float32(3.0))))) / Float32(Float32(-1.0) * fma(t_3, t_2, Float32((t_3 ^ Float32(2.0)) + (t_2 ^ Float32(2.0))))))
end

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 96.9%

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

7. Taylor expanded in s around 0

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

9. Applied rewrites 97.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{s \cdot \left({\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{3} - {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right), \log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right), {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{2} + {\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{2}\right)}} \]

11. Applied rewrites 91.7%

    \[\leadsto -1 \cdot \frac{s \cdot \left(e^{\log \log \left({\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3} - 1\right) \cdot 3} - {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{3}\right)}{\mathsf{fma}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right), \log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right), {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{2} + {\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{2}\right)} \]

12. Final simplification 91.7%

    \[\leadsto \frac{s \cdot \left(e^{\log \log \left({\left(\mathsf{fma}\left(u, \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{\pi}{s}\right)}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3} - 1\right) \cdot 3} + -1 \cdot {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{3}\right)}{-1 \cdot \mathsf{fma}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right), \log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right), {\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{2}}\right)\right)}^{2} + {\log \left(\frac{1}{{\left(\mathsf{fma}\left(u, \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}, \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{3}} - 1\right)}^{2}\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))