Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 16.6s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u around inf

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. lower-+.f32 N/A

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

5. Applied rewrites 98.7%

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-neg.f32 N/A

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

   8. remove-double-div N/A

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

   9. lower-*.f32 98.7

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

8. Applied rewrites 98.7%

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

9. Final simplification 98.7%

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

Alternative 4: 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 98.7%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 97.4%

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

6. Final simplification 97.4%

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

Alternative 5: 24.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in s around -inf

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f32 N/A

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

9. Applied rewrites 96.4%

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

10. Taylor expanded in s around inf

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

12. Applied rewrites 25.0%

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

13. Final simplification 25.0%

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

Alternative 6: 24.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\mathsf{fma}\left(0.5, u \cdot \pi, \pi \cdot -0.25\right)}{s}, 1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (fma -4.0 (/ (fma 0.5 (* u PI) (* PI -0.25)) s) 1.0))))

C

float code(float u, float s) {
	return -s * logf(fmaf(-4.0f, (fmaf(0.5f, (u * ((float) M_PI)), (((float) M_PI) * -0.25f)) / s), 1.0f));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log(fma(Float32(-4.0), Float32(fma(Float32(0.5), Float32(u * Float32(pi)), Float32(Float32(pi) * Float32(-0.25))) / s), Float32(1.0))))
end

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 10.0%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{0.5}} - 1\right) \]

6. Taylor expanded in s around inf

   \[\leadsto \left(\mathsf{neg}\left(s\right)\right) \cdot \log \color{blue}{\left(\left(1 + -1 \cdot \frac{-8 \cdot {\left(u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)\right)}^{2} + -4 \cdot \left(\frac{-1}{8} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{{s}^{2}}\right) - 4 \cdot \frac{u \cdot \left(\frac{1}{4} \cdot \mathsf{PI}\left(\right) - \frac{-1}{4} \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{4} \cdot \mathsf{PI}\left(\right)}{s}\right)} \]
7. Step-by-step derivation

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

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

   2. metadata-eval N/A

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

   3. lower-+.f32 N/A

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

8. Applied rewrites 14.8%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(1 - \frac{\mathsf{fma}\left(-8, \mathsf{fma}\left(u, \pi \cdot 0.5, \pi \cdot -0.25\right) \cdot \mathsf{fma}\left(u, \pi \cdot 0.5, \pi \cdot -0.25\right), 0\right)}{s \cdot s}\right) + -4 \cdot \frac{\mathsf{fma}\left(u, \pi \cdot 0.5, \pi \cdot -0.25\right)}{s}\right)} \]

9. Taylor expanded in s around inf

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

11. Applied rewrites 25.0%

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

12. Final simplification 25.0%

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

Alternative 7: 14.0% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \frac{s \cdot s}{-s} \cdot \frac{\pi}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in u around 0

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

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 11.1

      \[\leadsto \left(-s\right) \cdot \frac{\color{blue}{\pi}}{s} \]

5. Applied rewrites 11.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\frac{\pi}{s}} \]
6. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - s\right)} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   3. flip-- N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - s \cdot s}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{0 - \color{blue}{s \cdot s}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   6. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(s \cdot s\right)}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{s \cdot s}\right)}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   8. distribute-rgt-neg-out N/A

      \[\leadsto \frac{\color{blue}{s \cdot \left(\mathsf{neg}\left(s\right)\right)}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   9. lift-neg.f32 N/A

      \[\leadsto \frac{s \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{\color{blue}{s \cdot \left(\mathsf{neg}\left(s\right)\right)}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   11. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{s \cdot \left(\mathsf{neg}\left(s\right)\right)}{0 + s}} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   12. lift-*.f32 N/A

       \[\leadsto \frac{\color{blue}{s \cdot \left(\mathsf{neg}\left(s\right)\right)}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   13. lift-neg.f32 N/A

       \[\leadsto \frac{s \cdot \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   14. distribute-rgt-neg-out N/A

       \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(s \cdot s\right)}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   15. lift-*.f32 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{s \cdot s}\right)}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   16. lower-neg.f32 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(s \cdot s\right)}}{0 + s} \cdot \frac{\mathsf{PI}\left(\right)}{s} \]

   17. lower-+.f32 13.5

       \[\leadsto \frac{-s \cdot s}{\color{blue}{0 + s}} \cdot \frac{\pi}{s} \]

7. Applied rewrites 13.5%

   \[\leadsto \color{blue}{\frac{-s \cdot s}{0 + s}} \cdot \frac{\pi}{s} \]

8. Final simplification 13.5%

   \[\leadsto \frac{s \cdot s}{-s} \cdot \frac{\pi}{s} \]
9. Add Preprocessing

Alternative 8: 11.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in s around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

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

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 11.7

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

5. Applied rewrites 11.7%

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

6. Final simplification 11.7%

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

Alternative 9: 11.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 11.7

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

5. Applied rewrites 11.7%

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

Alternative 10: 11.5% 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 98.7%

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
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 11.1

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

5. Applied rewrites 11.1%

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

Alternative 11: 10.3% 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 98.7%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in s around inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. distribute-rgt-out N/A

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

   4. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{\frac{1}{2} + u \cdot \color{blue}{0}} - 1\right)\right)\right) \]

   5. mul0-rgt N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{\frac{1}{2} + \color{blue}{0}} - 1\right)\right)\right) \]

   6. metadata-eval N/A

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

   7. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \left(\color{blue}{2} - 1\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\log \color{blue}{1}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto s \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right) \]

   10. metadata-eval N/A

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

   11. lower-*.f32 10.0

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

9. Applied rewrites 10.0%

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

10. Taylor expanded in s around 0

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

12. Applied rewrites 10.0%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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))))