Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 100.0%

Time: 22.6s

Alternatives: 13

Speedup: 0.4×

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: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \langle \left( \left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (cast
  (!
   :precision
   binary64
   (*
    (- s)
    (log
     (+
      -1.0
      (/
       1.0
       (-
        (/ u (+ 1.0 (exp (/ PI (- s)))))
        (/ (+ u -1.0) (+ 1.0 (exp (/ PI s))))))))))))

C

float code(float u, float s) {
	double tmp = -((double) s) * log((-1.0 + (1.0 / ((((double) u) / (1.0 + exp((((double) M_PI) / -((double) s))))) - ((((double) u) + -1.0) / (1.0 + exp((((double) M_PI) / ((double) s)))))))));
	return (float) tmp;
}

Julia

function code(u, s)
	tmp = Float64(Float64(-Float64(s)) * log(Float64(-1.0 + Float64(1.0 / Float64(Float64(Float64(u) / Float64(1.0 + exp(Float64(pi / Float64(-Float64(s)))))) - Float64(Float64(Float64(u) + -1.0) / Float64(1.0 + exp(Float64(pi / Float64(s))))))))))
	return Float32(tmp)
end

MATLAB

function tmp_2 = code(u, s)
	tmp = -s * log((-1.0 + (1.0 / ((double(u) / (1.0 + exp((pi / -s)))) - ((double(u) + -1.0) / (1.0 + exp((pi / double(s)))))))));
	tmp_2 = single(tmp);
end

Derivation

1. Initial program 100.0%

   \[\langle s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right) \rangle_{\text{binary64}} \]

2. Final simplification 100.0%

   \[\leadsto \langle \left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}}\right) \rangle_{\text{binary64}} \]

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \langle \left( \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}}\right) \right)_{\text{binary64}} \rangle_{\text{binary32}} \cdot \left(-s\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (cast
   (!
    :precision
    binary64
    (log
     (+
      -1.0
      (/
       1.0
       (-
        (/ u (+ 1.0 (exp (/ PI (- s)))))
        (/ (+ u -1.0) (+ 1.0 (exp (/ PI s))))))))))
  (- s)))

C

float code(float u, float s) {
	double tmp = log((-1.0 + (1.0 / ((((double) u) / (1.0 + exp((((double) M_PI) / -((double) s))))) - ((((double) u) + -1.0) / (1.0 + exp((((double) M_PI) / ((double) s)))))))));
	return ((float) tmp) * -s;
}

Julia

function code(u, s)
	tmp = log(Float64(-1.0 + Float64(1.0 / Float64(Float64(Float64(u) / Float64(1.0 + exp(Float64(pi / Float64(-Float64(s)))))) - Float64(Float64(Float64(u) + -1.0) / Float64(1.0 + exp(Float64(pi / Float64(s)))))))))
	return Float32(Float32(tmp) * Float32(-s))
end

MATLAB

function tmp_2 = code(u, s)
	tmp = log((-1.0 + (1.0 / ((double(u) / (1.0 + exp((pi / -s)))) - ((double(u) + -1.0) / (1.0 + exp((pi / double(s)))))))));
	tmp_2 = single((single(tmp) * double(-s)));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

   1. rewrite-binary32/binary64 99.1%

      \[\leadsto \color{blue}{s \cdot \left(-\langle \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}} + -1\right) \rangle_{\text{binary64}}\right)} \]

5. Applied rewrite-once 99.1%

   \[\leadsto s \cdot \left(-\color{blue}{\langle \color{blue}{\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \rangle_{\text{binary64}}}\right) \]

6. Final simplification 99.1%

   \[\leadsto \langle \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}}\right) \rangle_{\text{binary64}} \cdot \left(-s\right) \]

Alternative 3: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = (u / (single(1.0) + exp((single(pi) / -s)))) - ((u + single(-1.0)) / (single(1.0) + exp((single(pi) / s))));
	tmp = s * -log(((single(1.0) - (t_0 ^ single(-2.0))) / (single(-1.0) + (single(-1.0) / t_0))));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around 0 98.9%

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

   1. associate-*r* 98.9%

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

   2. neg-mul-1 98.9%

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

   3. sub-neg 98.9%

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

6. Simplified 98.9%

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

   1. +-commutative 98.9%

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

   2. flip-+ 98.8%

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

8. Applied egg-rr 98.9%

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

9. Final simplification 98.9%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	float t_0 = 1.0f + 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 / t_0))))));
}

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + 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 / t_0))))));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around 0 98.9%

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

   1. associate-*r* 98.9%

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

   2. neg-mul-1 98.9%

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

   3. sub-neg 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 5: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

   1. clear-num 98.8%

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

   2. inv-pow 98.8%

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

5. Applied egg-rr 98.8%

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

   1. unpow-1 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 6: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf((-1.0f + (1.0f / ((u / (1.0f + expf((((float) M_PI) / -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) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) - Float32(Float32(u + 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) + exp((single(pi) / -s)))) - ((u + single(-1.0)) / (single(1.0) + exp((single(pi) / s))))))));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 7: 24.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around 0 98.9%

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

   1. associate-*r* 98.9%

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

   2. neg-mul-1 98.9%

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

   3. sub-neg 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in s around -inf 24.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}\right)} \]
8. Step-by-step derivation

   1. +-commutative 24.9%

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

   2. fma-def 24.9%

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

9. Simplified 24.9%

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

10. Final simplification 24.9%

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

Alternative 8: 24.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around 0 98.9%

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

   1. associate-*r* 98.9%

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

   2. neg-mul-1 98.9%

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

   3. sub-neg 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in s around inf 24.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}\right)} \]
8. Step-by-step derivation

   1. *-commutative 24.9%

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

   2. *-commutative 24.9%

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

   3. associate-*r* 24.9%

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

   4. *-commutative 24.9%

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

   5. distribute-rgt-in 24.9%

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

   6. fma-def 24.9%

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

   7. *-commutative 24.9%

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

   8. cancel-sign-sub-inv 24.9%

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

   9. associate-*r* 24.9%

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

   10. *-commutative 24.9%

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

   11. distribute-rgt-out 24.9%

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

   12. sub-neg 24.9%

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

9. Simplified 24.9%

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

10. Final simplification 24.9%

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

Alternative 9: 24.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(1.0) + ((single(pi) * (single(-0.25) + (u * single(0.5)))) * (single(-4.0) / s))));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around 0 98.9%

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

   1. associate-*r* 98.9%

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

   2. neg-mul-1 98.9%

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

   3. sub-neg 98.9%

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

6. Simplified 98.9%

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

   1. +-commutative 98.9%

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

   2. flip-+ 98.8%

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in s around inf 24.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -1 \cdot \frac{5.333333333333333 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right) - 1.3333333333333333 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right)}{s}\right)} \]

11. Simplified 24.8%

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

12. Final simplification 24.8%

    \[\leadsto s \cdot \left(-\log \left(1 + \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right) \cdot \frac{-4}{s}\right)\right) \]

Alternative 10: 24.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(1.0) - ((single(4.0) * (single(pi) * (single(-0.25) + (u * single(0.5))))) / s)));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around 0 98.9%

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

   1. associate-*r* 98.9%

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

   2. neg-mul-1 98.9%

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

   3. sub-neg 98.9%

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

6. Simplified 98.9%

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

   1. +-commutative 98.9%

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

   2. flip-+ 98.8%

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in s around inf 24.9%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + -1 \cdot \frac{5.333333333333333 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right) - 1.3333333333333333 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right)}{s}\right)} \]

11. Simplified 24.9%

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

12. Final simplification 24.9%

    \[\leadsto s \cdot \left(-\log \left(1 - \frac{4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right)}{s}\right)\right) \]

Alternative 11: 11.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(\pi \cdot \left(u \cdot 0.25 - \left(0.25 + u \cdot -0.25\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (single(pi) * ((u * single(0.25)) - (single(0.25) + (u * single(-0.25)))));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

   1. add-exp-log_binary32 95.1%

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

5. Applied rewrite-once 95.1%

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

6. Taylor expanded in s around inf 11.1%

   \[\leadsto s \cdot \left(-\color{blue}{-4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)}{s}}\right) \]
7. Step-by-step derivation

   1. +-commutative 11.1%

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

   2. *-commutative 11.1%

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

   3. associate-*r* 11.1%

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

   4. *-commutative 11.1%

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

   5. *-commutative 11.1%

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

   6. associate-*r* 11.1%

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

   7. +-commutative 11.1%

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

   8. distribute-lft-in 11.1%

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

   9. *-commutative 11.1%

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

   10. distribute-rgt-out-- 11.1%

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

   11. *-commutative 11.1%

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

   12. fma-def 11.1%

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

8. Simplified 11.1%

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

9. Taylor expanded in s around 0 11.1%

   \[\leadsto \color{blue}{4 \cdot \left(\pi \cdot \left(0.25 \cdot u - \left(0.25 + -0.25 \cdot u\right)\right)\right)} \]

10. Final simplification 11.1%

    \[\leadsto 4 \cdot \left(\pi \cdot \left(u \cdot 0.25 - \left(0.25 + u \cdot -0.25\right)\right)\right) \]

Alternative 12: 11.5% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(pi) * (single(-1.0) + (u * single(2.0)));
end

Derivation

1. Initial program 98.9%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around -inf 11.1%

   \[\leadsto s \cdot \left(-\color{blue}{4 \cdot \frac{-0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \pi + 0.25 \cdot \left(u \cdot \pi\right)\right)}{s}}\right) \]
5. Step-by-step derivation

   1. associate-*r/ 11.1%

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

   2. associate--r+ 11.1%

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

   3. cancel-sign-sub-inv 11.1%

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

   4. cancel-sign-sub-inv 11.1%

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

   5. metadata-eval 11.1%

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

   6. associate-*r* 11.1%

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

   7. distribute-rgt-out 11.1%

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

   8. *-commutative 11.1%

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

   9. metadata-eval 11.1%

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

   10. associate-*r* 11.1%

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

   11. *-commutative 11.1%

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

   12. *-commutative 11.1%

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

6. Simplified 11.1%

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

7. Taylor expanded in u around 0 11.1%

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

   1. associate-*r* 11.1%

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

   2. distribute-rgt-out 11.1%

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

   3. *-commutative 11.1%

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

9. Simplified 11.1%

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

10. Final simplification 11.1%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]

Alternative 13: 11.3% accurate, 7.2× 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%

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

3. Simplified 98.8%

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

4. Taylor expanded in u around 0 10.8%

   \[\leadsto \color{blue}{-1 \cdot \pi} \]
5. Step-by-step derivation

   1. neg-mul-1 10.8%

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

6. Simplified 10.8%

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

7. Final simplification 10.8%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023297 
(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))))