Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 22.7s

Alternatives: 8

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 86.4%

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

   1. +-commutative 86.4%

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

7. Simplified 86.4%

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

8. Taylor expanded in s around 0 97.6%

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

   1. associate-*r* 97.6%

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

   2. neg-mul-1 97.6%

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

   3. sub-neg 97.6%

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

   4. mul-1-neg 97.6%

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

   5. distribute-frac-neg2 97.6%

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

   6. metadata-eval 97.6%

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

10. Simplified 97.6%

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

11. Taylor expanded in u around 0 97.7%

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

    1. associate-+r+ 97.7%

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

    2. +-commutative 97.7%

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

    3. associate-+l+ 97.7%

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

    4. mul-1-neg 97.7%

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

    5. distribute-frac-neg2 97.7%

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

    6. mul-1-neg 97.7%

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

    7. sub-neg 97.7%

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

13. Simplified 97.7%

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

14. Final simplification 97.7%

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

Alternative 3: 76.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 86.4%

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

   1. +-commutative 86.4%

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

7. Simplified 86.4%

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

8. Taylor expanded in s around 0 97.6%

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

   1. associate-*r* 97.6%

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

   2. neg-mul-1 97.6%

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

   3. sub-neg 97.6%

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

   4. mul-1-neg 97.6%

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

   5. distribute-frac-neg2 97.6%

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

   6. metadata-eval 97.6%

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

10. Simplified 97.6%

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

11. Taylor expanded in u around 0 77.3%

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

    1. mul-1-neg 77.3%

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

    2. distribute-frac-neg2 77.3%

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

13. Simplified 77.3%

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

14. Final simplification 77.3%

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

Alternative 4: 37.2% 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 86.4%

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

   1. +-commutative 86.4%

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

7. Simplified 86.4%

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

8. Taylor expanded in s around 0 97.6%

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

   1. associate-*r* 97.6%

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

   2. neg-mul-1 97.6%

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

   3. sub-neg 97.6%

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

   4. mul-1-neg 97.6%

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

   5. distribute-frac-neg2 97.6%

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

   6. metadata-eval 97.6%

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

10. Simplified 97.6%

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

11. Taylor expanded in s around inf 37.3%

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

    1. sub-neg 37.3%

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

    2. associate-*r/ 37.3%

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

    3. metadata-eval 37.3%

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

    4. metadata-eval 37.3%

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

13. Simplified 37.3%

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

14. Final simplification 37.3%

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

Alternative 5: 25.2% 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 86.4%

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

   1. +-commutative 86.4%

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

7. Simplified 86.4%

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

8. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. neg-mul-1 25.1%

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

   3. log1p-define 25.1%

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

10. Simplified 25.1%

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

11. Final simplification 25.1%

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

Alternative 6: 11.7% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. exp-prod 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in s around inf 11.4%

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

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. distribute-rgt-out-- 11.4%

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

   4. metadata-eval 11.4%

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

   5. *-commutative 11.4%

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

   6. metadata-eval 11.4%

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

   7. +-commutative 11.4%

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

   8. associate-*r* 11.4%

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

   9. *-commutative 11.4%

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

   10. distribute-rgt-out 11.4%

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

9. Simplified 11.4%

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

10. Final simplification 11.4%

    \[\leadsto 4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right) \]
11. Add Preprocessing

Alternative 7: 11.7% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. exp-prod 98.8%

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

6. Applied egg-rr 98.8%

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

   1. add-cube-cbrt 97.5%

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

   2. pow3 97.6%

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

8. Applied egg-rr 97.6%

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

9. Taylor expanded in s around inf 11.4%

   \[\leadsto {\left(\sqrt[3]{\color{blue}{4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right)}}\right)}^{3} \]
10. Step-by-step derivation

    1. associate--r+ 11.4%

       \[\leadsto {\left(\sqrt[3]{4 \cdot \color{blue}{\left(\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) - 0.25 \cdot \pi\right)}}\right)}^{3} \]

    2. cancel-sign-sub-inv 11.4%

       \[\leadsto {\left(\sqrt[3]{4 \cdot \color{blue}{\left(\left(0.25 \cdot \left(u \cdot \pi\right) - -0.25 \cdot \left(u \cdot \pi\right)\right) + \left(-0.25\right) \cdot \pi\right)}}\right)}^{3} \]

    3. distribute-rgt-out-- 11.4%

       \[\leadsto {\left(\sqrt[3]{4 \cdot \left(\color{blue}{\left(u \cdot \pi\right) \cdot \left(0.25 - -0.25\right)} + \left(-0.25\right) \cdot \pi\right)}\right)}^{3} \]

    4. metadata-eval 11.4%

       \[\leadsto {\left(\sqrt[3]{4 \cdot \left(\left(u \cdot \pi\right) \cdot \color{blue}{0.5} + \left(-0.25\right) \cdot \pi\right)}\right)}^{3} \]

    5. *-commutative 11.4%

       \[\leadsto {\left(\sqrt[3]{4 \cdot \left(\color{blue}{\left(\pi \cdot u\right)} \cdot 0.5 + \left(-0.25\right) \cdot \pi\right)}\right)}^{3} \]

    6. metadata-eval 11.4%

       \[\leadsto {\left(\sqrt[3]{4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{-0.25} \cdot \pi\right)}\right)}^{3} \]

    7. *-commutative 11.4%

       \[\leadsto {\left(\sqrt[3]{4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \color{blue}{\pi \cdot -0.25}\right)}\right)}^{3} \]

11. Simplified 11.4%

    \[\leadsto {\left(\sqrt[3]{\color{blue}{4 \cdot \left(\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25\right)}}\right)}^{3} \]

12. Taylor expanded in u around inf 11.4%

    \[\leadsto \color{blue}{u \cdot \left(-0.25 \cdot \frac{\pi \cdot {\left(\sqrt[3]{4}\right)}^{3}}{u} + 0.5 \cdot \left(\pi \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)\right)} \]
13. Step-by-step derivation

    1. associate-*r/ 11.4%

       \[\leadsto u \cdot \left(\color{blue}{\frac{-0.25 \cdot \left(\pi \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)}{u}} + 0.5 \cdot \left(\pi \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)\right) \]

    2. rem-cube-cbrt 11.4%

       \[\leadsto u \cdot \left(\frac{-0.25 \cdot \left(\pi \cdot \color{blue}{4}\right)}{u} + 0.5 \cdot \left(\pi \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)\right) \]

    3. *-commutative 11.4%

       \[\leadsto u \cdot \left(\frac{-0.25 \cdot \color{blue}{\left(4 \cdot \pi\right)}}{u} + 0.5 \cdot \left(\pi \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)\right) \]

    4. associate-*r* 11.4%

       \[\leadsto u \cdot \left(\frac{\color{blue}{\left(-0.25 \cdot 4\right) \cdot \pi}}{u} + 0.5 \cdot \left(\pi \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)\right) \]

    5. metadata-eval 11.4%

       \[\leadsto u \cdot \left(\frac{\color{blue}{-1} \cdot \pi}{u} + 0.5 \cdot \left(\pi \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)\right) \]

    6. neg-mul-1 11.4%

       \[\leadsto u \cdot \left(\frac{\color{blue}{-\pi}}{u} + 0.5 \cdot \left(\pi \cdot {\left(\sqrt[3]{4}\right)}^{3}\right)\right) \]

    7. rem-cube-cbrt 11.4%

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

    8. *-commutative 11.4%

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

    9. associate-*r* 11.4%

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

    10. metadata-eval 11.4%

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

14. Simplified 11.4%

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

15. Final simplification 11.4%

    \[\leadsto u \cdot \left(\pi \cdot 2 - \frac{\pi}{u}\right) \]
16. Add Preprocessing

Alternative 8: 11.4% 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 11.2%

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

   1. neg-mul-1 11.2%

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

7. Simplified 11.2%

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

8. Final simplification 11.2%

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

Reproduce

herbie shell --seed 2024079 
(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))))