Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.9%

Time: 16.7s

Alternatives: 10

Speedup: 1.3×

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: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

   1. add-exp-log 99.1%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 3: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(-log1p(Float32(Float32(-1.0) + Float32(Float32(Float32(1.0) / u) + expm1(Float32(Float32(Float32(pi) / Float32(-s)) - log(u))))))))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 87.8%

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

   1. +-commutative 87.8%

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

7. Simplified 87.8%

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

8. Taylor expanded in s around 0 97.9%

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

   1. associate-*r* 97.9%

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

   2. neg-mul-1 97.9%

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

   3. associate--l+ 97.9%

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

   4. mul-1-neg 97.9%

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

   5. distribute-frac-neg2 97.9%

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

10. Simplified 97.9%

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

    1. log1p-expm1-u 97.9%

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

    2. expm1-undefine 97.9%

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

    3. add-exp-log 97.9%

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

    4. add-exp-log 97.9%

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

    5. expm1-define 97.9%

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

    6. log-div 97.9%

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

    7. add-log-exp 97.9%

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

    8. distribute-frac-neg2 97.9%

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

12. Applied egg-rr 97.9%

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

13. Final simplification 97.9%

    \[\leadsto s \cdot \left(-\mathsf{log1p}\left(-1 + \left(\frac{1}{u} + \mathsf{expm1}\left(\frac{\pi}{-s} - \log u\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 4: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 87.8%

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

   1. +-commutative 87.8%

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

7. Simplified 87.8%

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

8. Taylor expanded in s around 0 97.9%

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

   1. associate-*r* 97.9%

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

   2. neg-mul-1 97.9%

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

   3. associate--l+ 97.9%

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

   4. mul-1-neg 97.9%

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

   5. distribute-frac-neg2 97.9%

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

10. Simplified 97.9%

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

11. Final simplification 97.9%

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

Alternative 5: 37.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf((-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)))
end function

Julia

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 87.8%

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

   1. +-commutative 87.8%

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

7. Simplified 87.8%

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

8. Taylor expanded in s around 0 97.9%

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

   1. associate-*r* 97.9%

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

   2. neg-mul-1 97.9%

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

   3. associate--l+ 97.9%

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

   4. mul-1-neg 97.9%

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

   5. distribute-frac-neg2 97.9%

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

10. Simplified 97.9%

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

11. Taylor expanded in s around inf 36.8%

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

    1. associate-*r* 36.8%

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

    2. neg-mul-1 36.8%

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

    3. sub-neg 36.8%

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

    4. associate-*r/ 36.8%

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

    5. metadata-eval 36.8%

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

    6. metadata-eval 36.8%

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

13. Simplified 36.8%

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

14. Final simplification 36.8%

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

Alternative 6: 25.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* s (- (log1p (/ PI s)))))

C

float code(float u, float s) {
	return s * -log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 87.8%

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

   1. +-commutative 87.8%

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

7. Simplified 87.8%

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

8. Taylor expanded in u around 0 25.0%

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

   1. associate-*r* 25.0%

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

   2. neg-mul-1 25.0%

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

   3. log1p-define 25.0%

      \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

10. Simplified 25.0%

    \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

11. Final simplification 25.0%

    \[\leadsto s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]
12. Add Preprocessing

Alternative 7: 16.2% accurate, 28.9× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(\left(u \cdot -4\right) \cdot \left(\frac{1}{\frac{\pi}{s} - 2} - -0.5\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (* (* u -4.0) (- (/ 1.0 (- (/ PI s) 2.0)) -0.5))))

C

float code(float u, float s) {
	return s * ((u * -4.0f) * ((1.0f / ((((float) M_PI) / s) - 2.0f)) - -0.5f));
}

Julia

function code(u, s)
	return Float32(s * Float32(Float32(u * Float32(-4.0)) * Float32(Float32(Float32(1.0) / Float32(Float32(Float32(pi) / s) - Float32(2.0))) - Float32(-0.5))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * ((u * single(-4.0)) * ((single(1.0) / ((single(pi) / s) - single(2.0))) - single(-0.5)));
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 9.2%

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

6. Taylor expanded in s around inf 15.7%

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

   1. neg-mul-1 15.7%

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

   2. unsub-neg 15.7%

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

8. Simplified 15.7%

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

9. Taylor expanded in u around 0 16.4%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(-4 \cdot \left(u \cdot \left(\frac{1}{2 - \frac{\pi}{s}} - 0.5\right)\right)\right)} \]
10. Step-by-step derivation

    1. associate-*r* 16.4%

       \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{2 - \frac{\pi}{s}} - 0.5\right)\right)} \]

    2. sub-neg 16.4%

       \[\leadsto \left(-s\right) \cdot \left(\left(-4 \cdot u\right) \cdot \color{blue}{\left(\frac{1}{2 - \frac{\pi}{s}} + \left(-0.5\right)\right)}\right) \]

    3. metadata-eval 16.4%

       \[\leadsto \left(-s\right) \cdot \left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{2 - \frac{\pi}{s}} + \color{blue}{-0.5}\right)\right) \]

11. Simplified 16.4%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\left(-4 \cdot u\right) \cdot \left(\frac{1}{2 - \frac{\pi}{s}} + -0.5\right)\right)} \]

12. Final simplification 16.4%

    \[\leadsto s \cdot \left(\left(u \cdot -4\right) \cdot \left(\frac{1}{\frac{\pi}{s} - 2} - -0.5\right)\right) \]
13. Add Preprocessing

Alternative 8: 16.2% accurate, 86.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 9.2%

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

6. Taylor expanded in s around inf 15.7%

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

   1. neg-mul-1 15.7%

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

   2. unsub-neg 15.7%

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

8. Simplified 15.7%

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

9. Taylor expanded in s around 0 15.8%

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

    1. sub-neg 15.8%

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

    2. associate-*r/ 15.8%

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

    3. metadata-eval 15.8%

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

    4. metadata-eval 15.8%

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

11. Simplified 15.8%

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

12. Taylor expanded in u around 0 16.4%

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

13. Final simplification 16.4%

    \[\leadsto -2 \cdot \left(s \cdot u\right) \]
14. Add Preprocessing

Alternative 9: 16.2% accurate, 86.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 9.2%

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

6. Taylor expanded in s around inf 15.7%

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

   1. neg-mul-1 15.7%

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

   2. unsub-neg 15.7%

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

8. Simplified 15.7%

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

9. Taylor expanded in s around 0 15.8%

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

    1. sub-neg 15.8%

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

    2. associate-*r/ 15.8%

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

    3. metadata-eval 15.8%

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

    4. metadata-eval 15.8%

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

11. Simplified 15.8%

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

12. Taylor expanded in u around 0 16.4%

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

    1. associate-*r* 16.4%

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

    2. *-commutative 16.4%

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

14. Simplified 16.4%

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

15. Final simplification 16.4%

    \[\leadsto u \cdot \left(s \cdot -2\right) \]
16. Add Preprocessing

Alternative 10: 11.2% accurate, 216.5× 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u around 0 10.9%

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

   1. neg-mul-1 10.9%

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

7. Simplified 10.9%

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

8. Final simplification 10.9%

   \[\leadsto -\pi \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(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))))