Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 99.0%

Time: 22.0s

Alternatives: 20

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = (single(1.0) / (single(1.0) + t_0)) + ((u / (single(1.0) + exp((single(pi) / -s)))) + (u / (single(-1.0) - t_0)));
	tmp = s * log((((t_1 ^ single(-2.0)) + (single(1.0) + (single(1.0) / t_1))) / ((t_1 ^ single(-3.0)) + 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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	t_1 = (single(1.0) / (single(1.0) + t_0)) + ((u / (single(1.0) + exp((single(pi) / -s)))) + (u / (single(-1.0) - t_0)));
	tmp = s * log(((single(1.0) + (single(1.0) / t_1)) / ((t_1 ^ single(-2.0)) + 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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

7. Final simplification 99.0%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / s));
	tmp = -s * log((single(-1.0) + (single(1.0) / ((u / (single(1.0) + exp((single(pi) / -s)))) + ((single(1.0) / (single(1.0) + t_0)) + (u / (single(-1.0) - t_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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-+l+ N/A

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

   6. lower-+.f32 N/A

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

6. Applied rewrites 98.9%

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

7. Final simplification 98.9%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

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

Alternative 5: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(-1.0) / Float32(Float32(1.0) + Float32(fma(s, fma(s, Float32(s + Float32(pi)), Float32(Float32(0.5) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(0.16666666666666666) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))) / Float32(s * Float32(s * s)))))), u, Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))))))
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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

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 98.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 98.1%

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

11. Final simplification 98.1%

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

Alternative 6: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(-1.0) / Float32(Float32(1.0) + Float32(fma(s, fma(Float32(pi), s, Float32(Float32(0.5) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(0.16666666666666666) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))) / Float32(s * Float32(s * s)))))), u, Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))))))
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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

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 98.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 98.1%

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

11. Final simplification 98.1%

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

Alternative 7: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(-1.0) / Float32(Float32(1.0) + Float32(fma(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(0.16666666666666666), Float32(Float32(0.5) * Float32(s * Float32(Float32(pi) * Float32(pi))))) / Float32(s * Float32(s * s)))))), u, Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))))))
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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

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 98.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 98.1%

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

11. Final simplification 98.1%

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

Alternative 8: 98.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(-1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(-0.16666666666666666)) / Float32(s * s)) / s))))), u, Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))))))
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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

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 98.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 98.1%

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

11. Final simplification 98.1%

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

Alternative 9: 98.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / fma(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(Float32(-1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) / Float32(s * Float32(s * s)))))), u, Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))))))
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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

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 98.1%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 98.1%

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

11. Final simplification 98.1%

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

Alternative 10: 97.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. lower-+.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. mul-1-neg N/A

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

   10. lower-/.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f32 N/A

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

   14. distribute-neg-frac N/A

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

5. Applied rewrites 98.1%

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

6. Final simplification 98.1%

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

Alternative 11: 37.7% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (+
      (/ 1.0 (+ 1.0 (exp (/ PI s))))
      (*
       u
       (+
        (/ 1.0 (+ 1.0 1.0))
        (/
         1.0
         (-
          -1.0
          (+
           1.0
           (/
            (+
             PI
             (/
              (fma
               0.16666666666666666
               (/ (* PI (* PI PI)) s)
               (* PI (* PI 0.5)))
              s))
            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 / (1.0f + 1.0f)) + (1.0f / (-1.0f - (1.0f + ((((float) M_PI) + (fmaf(0.16666666666666666f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) / s), (((float) M_PI) * (((float) M_PI) * 0.5f))) / s)) / 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(Float32(1.0) / Float32(Float32(1.0) + Float32(1.0))) + Float32(Float32(1.0) / Float32(Float32(-1.0) - Float32(Float32(1.0) + Float32(Float32(Float32(pi) + Float32(fma(Float32(0.16666666666666666), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) / s), Float32(Float32(pi) * Float32(Float32(pi) * Float32(0.5)))) / s)) / s)))))))))))
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. Taylor expanded in s around inf

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

5. Applied rewrites 37.8%

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

6. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

8. Applied rewrites 37.8%

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

9. Final simplification 37.8%

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

Alternative 12: 37.7% accurate, 1.6× speedup

Math

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

FPCore

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

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) + Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(1.0))) + Float32(Float32(1.0) / Float32(Float32(-1.0) - Float32(Float32(1.0) - Float32(fma(Float32(-0.5), Float32(Float32(Float32(pi) * Float32(pi)) / s), Float32(-Float32(pi))) / s)))))))))))
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. Taylor expanded in s around inf

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

5. Applied rewrites 37.8%

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

6. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f32 N/A

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

   14. lower-PI.f32 37.8

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

8. Applied rewrites 37.8%

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

9. Final simplification 37.8%

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

Alternative 13: 37.7% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (+
      (/ 1.0 (+ 1.0 (exp (/ PI s))))
      (* u (+ (/ 1.0 (+ 1.0 1.0)) (/ 1.0 (- -1.0 (+ 1.0 (/ 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 / (1.0f + 1.0f)) + (1.0f / (-1.0f - (1.0f + (((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(Float32(1.0) / Float32(Float32(1.0) + Float32(1.0))) + Float32(Float32(1.0) / Float32(Float32(-1.0) - Float32(Float32(1.0) + Float32(Float32(pi) / 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) / (single(1.0) + single(1.0))) + (single(1.0) / (single(-1.0) - (single(1.0) + (single(pi) / s))))))))));
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. Taylor expanded in s around inf

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

5. Applied rewrites 37.8%

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

6. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-PI.f32 37.8

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

8. Applied rewrites 37.8%

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

9. Final simplification 37.8%

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

Alternative 14: 14.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\\ t_1 := 0 \cdot \left(t\_0 \cdot t\_0\right)\\ t_2 := -0.5 \cdot t\_1\\ \mathbf{if}\;s \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_2 \cdot t\_2}{s \cdot s} - \left(\pi \cdot \pi\right) \cdot \left(4 \cdot \left(u \cdot u\right)\right)}{\mathsf{fma}\left(-0.5, \frac{t\_1}{s}, t\_0 \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (fma u (* PI -0.5) (* PI 0.25)))
        (t_1 (* 0.0 (* t_0 t_0)))
        (t_2 (* -0.5 t_1)))
   (if (<= s 9.999999998199587e-24)
     0.0
     (/
      (- (/ (* t_2 t_2) (* s s)) (* (* PI PI) (* 4.0 (* u u))))
      (fma -0.5 (/ t_1 s) (* t_0 4.0))))))

C

float code(float u, float s) {
	float t_0 = fmaf(u, (((float) M_PI) * -0.5f), (((float) M_PI) * 0.25f));
	float t_1 = 0.0f * (t_0 * t_0);
	float t_2 = -0.5f * t_1;
	float tmp;
	if (s <= 9.999999998199587e-24f) {
		tmp = 0.0f;
	} else {
		tmp = (((t_2 * t_2) / (s * s)) - ((((float) M_PI) * ((float) M_PI)) * (4.0f * (u * u)))) / fmaf(-0.5f, (t_1 / s), (t_0 * 4.0f));
	}
	return tmp;
}

Julia

function code(u, s)
	t_0 = fma(u, Float32(Float32(pi) * Float32(-0.5)), Float32(Float32(pi) * Float32(0.25)))
	t_1 = Float32(Float32(0.0) * Float32(t_0 * t_0))
	t_2 = Float32(Float32(-0.5) * t_1)
	tmp = Float32(0.0)
	if (s <= Float32(9.999999998199587e-24))
		tmp = Float32(0.0);
	else
		tmp = Float32(Float32(Float32(Float32(t_2 * t_2) / Float32(s * s)) - Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(4.0) * Float32(u * u)))) / fma(Float32(-0.5), Float32(t_1 / s), Float32(t_0 * Float32(4.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 1e-23

   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. Taylor expanded in s around -inf

      \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}} \]

   5. Applied rewrites 4.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right), \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -16, \left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right)\right) \cdot 16\right)}{s}, \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -4\right)} \]

   6. Taylor expanded in s around 0

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{-16 \cdot {\left(\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + 16 \cdot {\left(\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{s}} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.0%

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

   9. Taylor expanded in s around 0

      \[\leadsto 0 \]
   10. Step-by-step derivation

   11. Applied rewrites 14.0%

       \[\leadsto 0 \]

   if 1e-23 < s

   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

      \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}} \]

   5. Applied rewrites 10.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right), \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -16, \left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right)\right) \cdot 16\right)}{s}, \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -4\right)} \]

   7. Applied rewrites 13.6%

      \[\leadsto \frac{\frac{\left(-0.5 \cdot \left(\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0\right)\right) \cdot \left(-0.5 \cdot \left(\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0\right)\right)}{s \cdot s} - \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot 16\right)}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0}{s}, 4 \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right)}} \]

   8. Taylor expanded in u around inf

      \[\leadsto \frac{\frac{\left(\frac{-1}{2} \cdot \left(\left(\mathsf{fma}\left(u, \mathsf{PI}\left(\right) \cdot \frac{-1}{2}, \mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{fma}\left(u, \mathsf{PI}\left(\right) \cdot \frac{-1}{2}, \mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right) \cdot 0\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\left(\mathsf{fma}\left(u, \mathsf{PI}\left(\right) \cdot \frac{-1}{2}, \mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{fma}\left(u, \mathsf{PI}\left(\right) \cdot \frac{-1}{2}, \mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right) \cdot 0\right)\right)}{s \cdot s} - 4 \cdot \left({u}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{\mathsf{fma}\left(\frac{-1}{2}, \frac{\left(\mathsf{fma}\left(u, \mathsf{PI}\left(\right) \cdot \frac{-1}{2}, \mathsf{PI}\left(\right) \cdot \frac{1}{4}\right) \cdot \mathsf{fma}\left(u, \mathsf{PI}\left(\right) \cdot \frac{-1}{2}, \mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right) \cdot 0}{s}, 4 \cdot \mathsf{fma}\left(u, \mathsf{PI}\left(\right) \cdot \frac{-1}{2}, \mathsf{PI}\left(\right) \cdot \frac{1}{4}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 15.3%

       \[\leadsto \frac{\frac{\left(-0.5 \cdot \left(\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0\right)\right) \cdot \left(-0.5 \cdot \left(\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0\right)\right)}{s \cdot s} - \left(4 \cdot \left(u \cdot u\right)\right) \cdot \left(\pi \cdot \pi\right)}{\mathsf{fma}\left(-0.5, \frac{\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0}{s}, 4 \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-0.5 \cdot \left(0 \cdot \left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right)\right)\right) \cdot \left(-0.5 \cdot \left(0 \cdot \left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right)\right)\right)}{s \cdot s} - \left(\pi \cdot \pi\right) \cdot \left(4 \cdot \left(u \cdot u\right)\right)}{\mathsf{fma}\left(-0.5, \frac{0 \cdot \left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right)}{s}, \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 11.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \left(u \cdot u\right) \cdot \left(\frac{0}{s} - \frac{\mathsf{fma}\left(\pi, -2, \frac{\pi}{u}\right)}{u}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (* u u) (- (/ 0.0 s) (/ (fma PI -2.0 (/ PI u)) u))))

C

float code(float u, float s) {
	return (u * u) * ((0.0f / s) - (fmaf(((float) M_PI), -2.0f, (((float) M_PI) / u)) / u));
}

Julia

function code(u, s)
	return Float32(Float32(u * u) * Float32(Float32(Float32(0.0) / s) - Float32(fma(Float32(pi), Float32(-2.0), Float32(Float32(pi) / u)) / u)))
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. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}} \]

5. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right), \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -16, \left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right)\right) \cdot 16\right)}{s}, \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -4\right)} \]

6. Taylor expanded in s around 0

   \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{-16 \cdot {\left(\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + 16 \cdot {\left(\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{s}} \]
7. Step-by-step derivation

8. Applied rewrites 10.3%

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

9. Taylor expanded in s around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 10.3%

    \[\leadsto 0 \]

12. Taylor expanded in u around -inf

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

14. Applied rewrites 11.1%

    \[\leadsto \left(u \cdot u\right) \cdot \color{blue}{\left(\frac{0}{s} - \frac{\mathsf{fma}\left(\pi, -2, \frac{\pi}{u}\right)}{u}\right)} \]
15. Add Preprocessing

Alternative 16: 11.6% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	t_0 = fma(Float32(Float32(pi) * u), Float32(-2.0), Float32(pi))
	return Float32(Float32(t_0 * t_0) / Float32(-t_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. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}} \]

5. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right), \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -16, \left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right)\right) \cdot 16\right)}{s}, \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -4\right)} \]

7. Applied rewrites 8.2%

   \[\leadsto \frac{\frac{\left(-0.5 \cdot \left(\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0\right)\right) \cdot \left(-0.5 \cdot \left(\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0\right)\right)}{s \cdot s} - \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot 16\right)}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0}{s}, 4 \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right)}} \]

9. Applied rewrites 11.1%

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

Alternative 17: 11.6% accurate, 36.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return fma(u, Float32(Float32(pi) * Float32(2.0)), Float32(-Float32(pi)))
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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in s around -inf

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

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

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

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

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

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f32 N/A

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

7. Applied rewrites 11.1%

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

Alternative 18: 11.6% accurate, 36.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(-fma(Float32(Float32(pi) * u), Float32(-2.0), Float32(pi)))
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. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}} \]

5. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right), \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -16, \left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right)\right) \cdot 16\right)}{s}, \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -4\right)} \]

7. Applied rewrites 8.2%

   \[\leadsto \frac{\frac{\left(-0.5 \cdot \left(\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0\right)\right) \cdot \left(-0.5 \cdot \left(\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0\right)\right)}{s \cdot s} - \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot 16\right)}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(\mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right) \cdot 0}{s}, 4 \cdot \mathsf{fma}\left(u, \pi \cdot -0.5, \pi \cdot 0.25\right)\right)}} \]

9. Applied rewrites 11.1%

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

Alternative 19: 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 98.9%

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

3. 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 11.0

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

5. Applied rewrites 11.0%

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

Alternative 20: 10.4% accurate, 510.0× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 0.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-4 \cdot \left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{2} \cdot \frac{-16 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -2 \cdot \left(-8 \cdot {\left(u \cdot \left(\frac{-1}{4} \cdot \mathsf{PI}\left(\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{s}} \]

5. Applied rewrites 8.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right), \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -16, \left(\mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right)\right) \cdot 16\right)}{s}, \mathsf{fma}\left(\pi, 0.25, u \cdot \left(\pi \cdot -0.5\right)\right) \cdot -4\right)} \]

6. Taylor expanded in s around 0

   \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{-16 \cdot {\left(\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + 16 \cdot {\left(\frac{-1}{2} \cdot \left(u \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{s}} \]
7. Step-by-step derivation

8. Applied rewrites 10.3%

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

9. Taylor expanded in s around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 10.3%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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))))