Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 24.9s

Alternatives: 9

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, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

   1. clear-num 98.9%

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

   2. inv-pow 98.9%

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

6. Applied egg-rr 98.9%

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

   1. unpow-1 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{1}{\frac{s}{\pi}}}}} + -1\right) \]
10. 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 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 98.7% accurate, 1.3× speedup

Math

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

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(s * Float32(-log(Float32(Float32(-1.0) + Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / 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 98.9%

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

3. Taylor expanded in s around inf 96.2%

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

   1. fma-def 96.2%

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

   2. unpow2 96.2%

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

   3. unpow2 96.2%

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

   4. times-frac 96.2%

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

   5. unpow2 96.2%

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

5. Simplified 96.2%

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

6. Taylor expanded in s around 0 98.2%

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

   1. sub-neg 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 4: 97.6% accurate, 2.0× speedup

Math

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

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(s * Float32(-log(Float32(Float32(Float32(1.0) / u) + Float32(Float32(-1.0) + Float32(exp(Float32(Float32(-Float32(pi)) / 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 85.9%

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

   1. +-commutative 85.9%

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

7. Simplified 85.9%

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

8. Taylor expanded in s around 0 96.7%

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

   1. associate-*r* 96.7%

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

   2. neg-mul-1 96.7%

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

   3. associate--l+ 96.7%

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

   4. associate-*r/ 96.7%

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

   5. neg-mul-1 96.7%

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

10. Simplified 96.7%

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

11. Final simplification 96.7%

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

Alternative 5: 75.5% accurate, 2.1× speedup

Math

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

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(s * Float32(-log(Float32(Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 85.9%

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

   1. +-commutative 85.9%

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

7. Simplified 85.9%

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

8. Taylor expanded in s around 0 96.7%

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

   1. associate-*r* 96.7%

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

   2. neg-mul-1 96.7%

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

   3. associate--l+ 96.7%

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

   4. associate-*r/ 96.7%

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

   5. neg-mul-1 96.7%

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

10. Simplified 96.7%

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

11. Taylor expanded in u around 0 75.6%

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

    1. metadata-eval 75.6%

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

    2. times-frac 75.6%

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

    3. *-lft-identity 75.6%

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

    4. neg-mul-1 75.6%

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

    5. *-lft-identity 75.6%

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

    6. neg-mul-1 75.6%

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

    7. times-frac 75.6%

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

    8. metadata-eval 75.6%

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

    9. neg-mul-1 75.6%

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

    10. distribute-neg-frac 75.6%

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

13. Simplified 75.6%

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

14. Final simplification 75.6%

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

Alternative 6: 36.9% accurate, 4.0× speedup

Math

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

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(s * Float32(-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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 85.9%

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

   1. +-commutative 85.9%

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

7. Simplified 85.9%

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

8. Taylor expanded in s around 0 96.7%

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

   1. associate-*r* 96.7%

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

   2. neg-mul-1 96.7%

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

   3. associate--l+ 96.7%

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

   4. associate-*r/ 96.7%

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

   5. neg-mul-1 96.7%

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

10. Simplified 96.7%

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

11. Taylor expanded in s around inf 37.1%

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

    1. sub-neg 37.1%

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

    2. associate-*r/ 37.1%

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

    3. metadata-eval 37.1%

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

    4. metadata-eval 37.1%

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

13. Simplified 37.1%

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

14. Final simplification 37.1%

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

Alternative 7: 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 85.9%

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

   1. +-commutative 85.9%

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

7. Simplified 85.9%

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

8. Taylor expanded in u around 0 25.4%

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

   1. associate-*r* 25.4%

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

   2. neg-mul-1 25.4%

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

   3. log1p-def 25.4%

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

10. Simplified 25.4%

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

11. Final simplification 25.4%

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

Alternative 8: 11.6% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around -inf 12.1%

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

   1. cancel-sign-sub-inv 12.1%

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

   2. distribute-rgt-out-- 12.1%

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

   3. metadata-eval 12.1%

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

   4. metadata-eval 12.1%

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

   5. *-commutative 12.1%

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

5. Simplified 12.1%

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

6. Final simplification 12.1%

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

Alternative 9: 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.9%

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

   1. neg-mul-1 11.9%

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

7. Simplified 11.9%

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

8. Final simplification 11.9%

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

Reproduce

herbie shell --seed 2024024 
(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))))