Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 15.9s

Alternatives: 11

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: 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + exp((single(pi) / s));
	tmp = -s * log(((single(1.0) / ((u / (single(1.0) + exp((single(pi) / -s)))) + (u * ((single(1.0) / (u * t_0)) - (single(1.0) / t_0))))) + 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. Taylor expanded in u around inf 98.8%

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

   1. *-commutative 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 2: 98.9% 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 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%

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

Alternative 3: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(-1.0) + (single(1.0) / ((u / (single(1.0) + exp((single(pi) / -s)))) + (single(1.0) / (single(1.0) + exp((single(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 u around 0 98.7%

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

6. Final simplification 98.7%

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

Alternative 4: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) + (exp((single(pi) / -s)) - 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 u around 0 98.7%

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

6. Taylor expanded in u around inf 97.7%

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

   1. associate--l+ 97.7%

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

   2. mul-1-neg 97.7%

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

   3. distribute-frac-neg2 97.7%

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

8. Simplified 97.7%

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

9. Taylor expanded in u around 0 97.8%

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

    1. neg-mul-1 97.8%

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

    2. unsub-neg 97.8%

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

    3. associate-*r/ 97.8%

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

    4. mul-1-neg 97.8%

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

11. Simplified 97.8%

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

12. Final simplification 97.8%

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

Alternative 5: 76.7% 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 u around 0 98.7%

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

6. Taylor expanded in u around inf 97.7%

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

   1. associate--l+ 97.7%

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

   2. mul-1-neg 97.7%

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

   3. distribute-frac-neg2 97.7%

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

8. Simplified 97.7%

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

9. Taylor expanded in u around 0 77.6%

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

    1. associate-*r/ 77.6%

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

    2. mul-1-neg 77.6%

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

11. Simplified 77.6%

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

12. Final simplification 77.6%

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

Alternative 6: 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 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 98.7%

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

6. Taylor expanded in u around inf 97.7%

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

   1. associate--l+ 97.7%

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

   2. mul-1-neg 97.7%

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

   3. distribute-frac-neg2 97.7%

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

8. Simplified 97.7%

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

9. Taylor expanded in s around inf 37.0%

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

    1. sub-neg 37.0%

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

    2. associate-*r/ 37.0%

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

    3. metadata-eval 37.0%

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

    4. metadata-eval 37.0%

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

11. Simplified 37.0%

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

12. Final simplification 37.0%

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

Alternative 7: 25.1% accurate, 4.1× speedup

Math

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

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(Float32(-s) * 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 25.1%

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

6. Taylor expanded in u around 0 25.5%

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

   1. associate-*r* 25.5%

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

   2. neg-mul-1 25.5%

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

   3. log1p-define 25.5%

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

8. Simplified 25.5%

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

Alternative 8: 12.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{{s}^{2}}{-\pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (s ^ single(2.0)) / -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 s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.5%

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

   1. associate-*r* 25.5%

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

   2. neg-mul-1 25.5%

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

   3. log1p-define 25.5%

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

8. Simplified 25.5%

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

9. Taylor expanded in s around 0 25.6%

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

    1. +-commutative 25.6%

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

    2. mul-1-neg 25.6%

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

    3. unsub-neg 25.6%

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

11. Simplified 25.6%

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

12. Taylor expanded in s around inf 12.6%

    \[\leadsto \color{blue}{-1 \cdot \frac{{s}^{2}}{\pi}} \]
13. Step-by-step derivation

    1. mul-1-neg 12.6%

       \[\leadsto \color{blue}{-\frac{{s}^{2}}{\pi}} \]

    2. distribute-neg-frac2 12.6%

       \[\leadsto \color{blue}{\frac{{s}^{2}}{-\pi}} \]

14. Simplified 12.6%

    \[\leadsto \color{blue}{\frac{{s}^{2}}{-\pi}} \]
15. Add Preprocessing

Alternative 9: 12.8% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * (s / -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 s around -inf 25.1%

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

6. Taylor expanded in u around 0 25.5%

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

   1. associate-*r* 25.5%

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

   2. neg-mul-1 25.5%

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

   3. log1p-define 25.5%

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

8. Simplified 25.5%

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

9. Taylor expanded in s around 0 25.6%

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

    1. +-commutative 25.6%

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

    2. mul-1-neg 25.6%

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

    3. unsub-neg 25.6%

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

11. Simplified 25.6%

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

12. Taylor expanded in s around inf 12.6%

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

13. Final simplification 12.6%

    \[\leadsto s \cdot \frac{s}{-\pi} \]
14. 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 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 11.4%

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

   1. neg-mul-1 11.4%

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

7. Simplified 11.4%

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

Alternative 11: 4.6% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ \pi \end{array} \]

FPCore

(FPCore (u s) :precision binary32 PI)

C

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

Julia

function code(u, s)
	return Float32(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 11.4%

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

   1. associate-*r/ 11.4%

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

   2. add-sqr-sqrt -0.0%

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

   3. sqrt-unprod 7.9%

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

   4. sqr-neg 7.9%

      \[\leadsto \frac{\sqrt{\color{blue}{s \cdot s}} \cdot \pi}{s} \]

   5. sqrt-unprod 4.6%

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

   6. add-sqr-sqrt 4.6%

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

7. Applied egg-rr 4.6%

   \[\leadsto \color{blue}{\frac{s \cdot \pi}{s}} \]
8. Step-by-step derivation

   1. associate-*r/ 4.6%

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

9. Simplified 4.6%

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

10. Taylor expanded in s around 0 4.6%

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

Reproduce

herbie shell --seed 2024139 
(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))))