Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 11.7s

Alternatives: 11

Speedup: 1.4×

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: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 37.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = log((single(-1.0) + (single(1.0) / ((u / single(2.0)) + ((single(1.0) - u) / (single(1.0) + exp((single(1.0) / (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. Step-by-step derivation

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 37.8%

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

   1. clear-num 37.8%

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

   2. inv-pow 37.8%

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

6. Applied egg-rr 37.8%

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

   1. unpow-1 37.8%

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

8. Simplified 37.8%

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

9. Final simplification 37.8%

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

Alternative 3: 37.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 37.8%

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

5. Final simplification 37.8%

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

Alternative 4: 37.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(-1 + \frac{2}{u}\right) \cdot \left(-s\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (log1p (expm1 (* (log (+ -1.0 (/ 2.0 u))) (- s)))))

C

float code(float u, float s) {
	return log1pf(expm1f((logf((-1.0f + (2.0f / u))) * -s)));
}

Julia

function code(u, s)
	return log1p(expm1(Float32(log(Float32(Float32(-1.0) + Float32(Float32(2.0) / u))) * 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. Step-by-step derivation

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 37.8%

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

5. Taylor expanded in s around inf 37.2%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + 1} + \frac{1 - u}{1 + \color{blue}{\left(1 + \left(\frac{\pi}{s} + 0.5 \cdot \frac{{\pi}^{2}}{{s}^{2}}\right)\right)}}} + -1\right)\right) \]
6. Step-by-step derivation

   1. unpow2 37.2%

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

7. Simplified 37.2%

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

8. Taylor expanded in s around 0 37.4%

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

   1. mul-1-neg 37.4%

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

   2. *-commutative 37.4%

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

   3. distribute-rgt-neg-in 37.4%

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

   4. sub-neg 37.4%

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

   5. associate-*r/ 37.4%

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

   6. metadata-eval 37.4%

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

   7. metadata-eval 37.4%

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

10. Simplified 37.4%

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

    1. log1p-expm1-u 37.4%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{2}{u} + -1\right) \cdot \left(-s\right)\right)\right)} \]

    2. +-commutative 37.4%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\log \color{blue}{\left(-1 + \frac{2}{u}\right)} \cdot \left(-s\right)\right)\right) \]

12. Applied egg-rr 37.4%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(-1 + \frac{2}{u}\right) \cdot \left(-s\right)\right)\right)} \]

13. Final simplification 37.4%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(-1 + \frac{2}{u}\right) \cdot \left(-s\right)\right)\right) \]

Alternative 5: 37.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{2}{u}\right)}^{3}}{\frac{2}{u} \cdot \frac{2}{u} + \left(1 + \frac{2}{u}\right)}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (/
    (+ -1.0 (pow (/ 2.0 u) 3.0))
    (+ (* (/ 2.0 u) (/ 2.0 u)) (+ 1.0 (/ 2.0 u)))))))

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -s * log((((-1.0e0) + ((2.0e0 / u) ** 3.0e0)) / (((2.0e0 / u) * (2.0e0 / u)) + (1.0e0 + (2.0e0 / u)))))
end function

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(-1.0) + (Float32(Float32(2.0) / u) ^ Float32(3.0))) / Float32(Float32(Float32(Float32(2.0) / u) * Float32(Float32(2.0) / u)) + Float32(Float32(1.0) + Float32(Float32(2.0) / u))))))
end

MATLAB

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 37.8%

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

5. Taylor expanded in s around inf 37.2%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + 1} + \frac{1 - u}{1 + \color{blue}{\left(1 + \left(\frac{\pi}{s} + 0.5 \cdot \frac{{\pi}^{2}}{{s}^{2}}\right)\right)}}} + -1\right)\right) \]
6. Step-by-step derivation

   1. unpow2 37.2%

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

7. Simplified 37.2%

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

8. Taylor expanded in s around 0 37.4%

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

   1. mul-1-neg 37.4%

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

   2. *-commutative 37.4%

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

   3. distribute-rgt-neg-in 37.4%

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

   4. sub-neg 37.4%

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

   5. associate-*r/ 37.4%

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

   6. metadata-eval 37.4%

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

   7. metadata-eval 37.4%

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

10. Simplified 37.4%

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

    1. flip3-+ 37.4%

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

    2. metadata-eval 37.4%

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

    3. metadata-eval 37.4%

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

12. Applied egg-rr 37.4%

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

13. Final simplification 37.4%

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

Alternative 6: 37.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \left(\frac{2}{u} + \frac{\frac{4}{u}}{u}\right)}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (/ (+ -1.0 (/ 8.0 (pow u 3.0))) (+ 1.0 (+ (/ 2.0 u) (/ (/ 4.0 u) u))))))))

C

float code(float u, float s) {
	return s * -logf(((-1.0f + (8.0f / powf(u, 3.0f))) / (1.0f + ((2.0f / u) + ((4.0f / u) / u)))));
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = s * -log((((-1.0e0) + (8.0e0 / (u ** 3.0e0))) / (1.0e0 + ((2.0e0 / u) + ((4.0e0 / u) / u)))))
end function

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(Float32(-1.0) + Float32(Float32(8.0) / (u ^ Float32(3.0)))) / Float32(Float32(1.0) + Float32(Float32(Float32(2.0) / u) + Float32(Float32(Float32(4.0) / u) / u)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log(((single(-1.0) + (single(8.0) / (u ^ single(3.0)))) / (single(1.0) + ((single(2.0) / u) + ((single(4.0) / u) / u)))));
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. Step-by-step derivation

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 37.8%

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

5. Taylor expanded in s around inf 37.2%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + 1} + \frac{1 - u}{1 + \color{blue}{\left(1 + \left(\frac{\pi}{s} + 0.5 \cdot \frac{{\pi}^{2}}{{s}^{2}}\right)\right)}}} + -1\right)\right) \]
6. Step-by-step derivation

   1. unpow2 37.2%

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

7. Simplified 37.2%

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

8. Taylor expanded in s around 0 37.4%

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

   1. mul-1-neg 37.4%

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

   2. *-commutative 37.4%

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

   3. distribute-rgt-neg-in 37.4%

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

   4. sub-neg 37.4%

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

   5. associate-*r/ 37.4%

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

   6. metadata-eval 37.4%

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

   7. metadata-eval 37.4%

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

10. Simplified 37.4%

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

    1. flip3-+ 37.4%

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

    2. metadata-eval 37.4%

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

    3. metadata-eval 37.4%

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

12. Applied egg-rr 37.4%

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

    1. +-commutative 37.4%

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

    2. cube-div 37.4%

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

    3. metadata-eval 37.4%

       \[\leadsto \log \left(\frac{-1 + \frac{\color{blue}{8}}{{u}^{3}}}{\frac{2}{u} \cdot \frac{2}{u} + \left(1 - \frac{2}{u} \cdot -1\right)}\right) \cdot \left(-s\right) \]

    4. +-commutative 37.4%

       \[\leadsto \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{\color{blue}{\left(1 - \frac{2}{u} \cdot -1\right) + \frac{2}{u} \cdot \frac{2}{u}}}\right) \cdot \left(-s\right) \]

    5. cancel-sign-sub-inv 37.4%

       \[\leadsto \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{\color{blue}{\left(1 + \left(-\frac{2}{u}\right) \cdot -1\right)} + \frac{2}{u} \cdot \frac{2}{u}}\right) \cdot \left(-s\right) \]

    6. associate-+l+ 37.4%

       \[\leadsto \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{\color{blue}{1 + \left(\left(-\frac{2}{u}\right) \cdot -1 + \frac{2}{u} \cdot \frac{2}{u}\right)}}\right) \cdot \left(-s\right) \]

    7. distribute-lft-neg-in 37.4%

       \[\leadsto \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \left(\color{blue}{\left(-\frac{2}{u} \cdot -1\right)} + \frac{2}{u} \cdot \frac{2}{u}\right)}\right) \cdot \left(-s\right) \]

    8. distribute-rgt-neg-in 37.4%

       \[\leadsto \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \left(\color{blue}{\frac{2}{u} \cdot \left(--1\right)} + \frac{2}{u} \cdot \frac{2}{u}\right)}\right) \cdot \left(-s\right) \]

    9. metadata-eval 37.4%

       \[\leadsto \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \left(\frac{2}{u} \cdot \color{blue}{1} + \frac{2}{u} \cdot \frac{2}{u}\right)}\right) \cdot \left(-s\right) \]

    10. *-rgt-identity 37.4%

        \[\leadsto \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \left(\color{blue}{\frac{2}{u}} + \frac{2}{u} \cdot \frac{2}{u}\right)}\right) \cdot \left(-s\right) \]

    11. associate-*l/ 37.4%

        \[\leadsto \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \left(\frac{2}{u} + \color{blue}{\frac{2 \cdot \frac{2}{u}}{u}}\right)}\right) \cdot \left(-s\right) \]

    12. associate-*r/ 37.4%

        \[\leadsto \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \left(\frac{2}{u} + \frac{\color{blue}{\frac{2 \cdot 2}{u}}}{u}\right)}\right) \cdot \left(-s\right) \]

    13. metadata-eval 37.4%

        \[\leadsto \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \left(\frac{2}{u} + \frac{\frac{\color{blue}{4}}{u}}{u}\right)}\right) \cdot \left(-s\right) \]

14. Simplified 37.4%

    \[\leadsto \log \color{blue}{\left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \left(\frac{2}{u} + \frac{\frac{4}{u}}{u}\right)}\right)} \cdot \left(-s\right) \]

15. Final simplification 37.4%

    \[\leadsto s \cdot \left(-\log \left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \left(\frac{2}{u} + \frac{\frac{4}{u}}{u}\right)}\right)\right) \]

Alternative 7: 37.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{-1 + \frac{\frac{4}{u}}{u}}{1 + \frac{2}{u}}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* (- s) (log (/ (+ -1.0 (/ (/ 4.0 u) u)) (+ 1.0 (/ 2.0 u))))))

C

float code(float u, float s) {
	return -s * logf(((-1.0f + ((4.0f / u) / u)) / (1.0f + (2.0f / u))));
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -s * log((((-1.0e0) + ((4.0e0 / u) / u)) / (1.0e0 + (2.0e0 / u))))
end function

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(-1.0) + Float32(Float32(Float32(4.0) / u) / u)) / Float32(Float32(1.0) + Float32(Float32(2.0) / u)))))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(-1.0) + ((single(4.0) / u) / u)) / (single(1.0) + (single(2.0) / u))));
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. Step-by-step derivation

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 37.8%

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

5. Taylor expanded in s around inf 37.2%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + 1} + \frac{1 - u}{1 + \color{blue}{\left(1 + \left(\frac{\pi}{s} + 0.5 \cdot \frac{{\pi}^{2}}{{s}^{2}}\right)\right)}}} + -1\right)\right) \]
6. Step-by-step derivation

   1. unpow2 37.2%

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

7. Simplified 37.2%

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

8. Taylor expanded in s around 0 37.4%

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

   1. mul-1-neg 37.4%

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

   2. *-commutative 37.4%

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

   3. distribute-rgt-neg-in 37.4%

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

   4. sub-neg 37.4%

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

   5. associate-*r/ 37.4%

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

   6. metadata-eval 37.4%

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

   7. metadata-eval 37.4%

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

10. Simplified 37.4%

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

    1. flip-+ 37.4%

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

    2. metadata-eval 37.4%

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

12. Applied egg-rr 37.4%

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

    1. sub-neg 37.4%

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

    2. associate-*l/ 37.4%

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

    3. associate-*r/ 37.4%

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

    4. metadata-eval 37.4%

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

    5. metadata-eval 37.4%

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

    6. sub-neg 37.4%

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

    7. metadata-eval 37.4%

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

14. Simplified 37.4%

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

15. Final simplification 37.4%

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

Alternative 8: 37.2% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = log(((-1.0e0) + (2.0e0 / u))) * -s
end function

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = log((single(-1.0) + (single(2.0) / u))) * -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. Step-by-step derivation

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 37.8%

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

5. Taylor expanded in s around inf 37.2%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + 1} + \frac{1 - u}{1 + \color{blue}{\left(1 + \left(\frac{\pi}{s} + 0.5 \cdot \frac{{\pi}^{2}}{{s}^{2}}\right)\right)}}} + -1\right)\right) \]
6. Step-by-step derivation

   1. unpow2 37.2%

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

7. Simplified 37.2%

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

8. Taylor expanded in s around 0 37.4%

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

   1. mul-1-neg 37.4%

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

   2. *-commutative 37.4%

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

   3. distribute-rgt-neg-in 37.4%

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

   4. sub-neg 37.4%

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

   5. associate-*r/ 37.4%

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

   6. metadata-eval 37.4%

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

   7. metadata-eval 37.4%

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

10. Simplified 37.4%

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

11. Final simplification 37.4%

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

Alternative 9: 37.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \left(-s\right) \cdot \log \left(\frac{2}{u}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log (/ 2.0 u))))

C

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

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = -s * log((2.0e0 / u))
end function

Julia

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(2.0) / u)))
end

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(2.0) / u));
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. Step-by-step derivation

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 37.8%

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

5. Taylor expanded in s around inf 37.2%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + 1} + \frac{1 - u}{1 + \color{blue}{\left(1 + \left(\frac{\pi}{s} + 0.5 \cdot \frac{{\pi}^{2}}{{s}^{2}}\right)\right)}}} + -1\right)\right) \]
6. Step-by-step derivation

   1. unpow2 37.2%

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

7. Simplified 37.2%

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

8. Taylor expanded in s around 0 37.4%

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

   1. mul-1-neg 37.4%

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

   2. *-commutative 37.4%

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

   3. distribute-rgt-neg-in 37.4%

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

   4. sub-neg 37.4%

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

   5. associate-*r/ 37.4%

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

   6. metadata-eval 37.4%

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

   7. metadata-eval 37.4%

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

10. Simplified 37.4%

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

11. Taylor expanded in u around 0 37.3%

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

12. Final simplification 37.3%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{2}{u}\right) \]

Alternative 10: 11.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = 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. Step-by-step derivation

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around inf 37.8%

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

5. Taylor expanded in u around 0 11.0%

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

6. Final simplification 11.0%

   \[\leadsto s \cdot \frac{-\pi}{s} \]

Alternative 11: 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 98.7%

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

   1. distribute-lft-neg-out 98.7%

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

   2. distribute-rgt-neg-in 98.7%

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

   3. sub-neg 98.7%

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

3. Simplified 98.8%

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

4. Taylor expanded in u around 0 11.0%

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

   1. mul-1-neg 11.0%

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

6. Simplified 11.0%

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

7. Final simplification 11.0%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023257 
(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))))