Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 22.0s

Alternatives: 6

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.4× speedup

Math

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

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(-Float32(pi)) / 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 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 76.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return s * -logf((-1.0f / (u * ((1.0f / (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(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) + Float32(Float32(-1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s))))))))))
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u around -inf 97.9%

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

   1. neg-sub0 97.9%

      \[\leadsto s \cdot \left(-\log \color{blue}{\left(0 - \left(\frac{1}{\left(\frac{1}{1 + e^{\frac{\pi}{s}}} - \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}}\right) \cdot u} + \left(1 + \frac{1}{\left(1 + e^{\frac{\pi}{s}}\right) \cdot \left(\left(\frac{1}{1 + e^{\frac{\pi}{s}}} - \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}}\right) \cdot \left(\left(\frac{1}{e^{\frac{\pi}{s}} + 1} - \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}}\right) \cdot {u}^{2}\right)\right)}\right)\right)\right)}\right) \]

   2. associate-+r+ 97.9%

      \[\leadsto s \cdot \left(-\log \left(0 - \color{blue}{\left(\left(\frac{1}{\left(\frac{1}{1 + e^{\frac{\pi}{s}}} - \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}}\right) \cdot u} + 1\right) + \frac{1}{\left(1 + e^{\frac{\pi}{s}}\right) \cdot \left(\left(\frac{1}{1 + e^{\frac{\pi}{s}}} - \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}}\right) \cdot \left(\left(\frac{1}{e^{\frac{\pi}{s}} + 1} - \frac{1}{1 + e^{-1 \cdot \frac{\pi}{s}}}\right) \cdot {u}^{2}\right)\right)}\right)}\right)\right) \]

6. Simplified 97.9%

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

7. Taylor expanded in u around 0 76.2%

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

   1. associate-/r* 76.2%

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

   2. +-commutative 76.2%

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

   3. distribute-frac-neg 76.2%

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

9. Simplified 76.2%

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

10. Taylor expanded in u around -inf -0.0%

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

    1. mul-1-neg -0.0%

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

    2. distribute-rgt-neg-in -0.0%

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

    3. +-commutative -0.0%

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

    4. log-rec -0.0%

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

    5. unsub-neg -0.0%

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

    6. sub-neg -0.0%

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

12. Simplified 76.4%

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

13. Final simplification 76.4%

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

Alternative 3: 25.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(1 + -4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \pi \cdot 0.25}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (- (log (+ 1.0 (* -4.0 (/ (- (* 0.25 (* u PI)) (* PI 0.25)) s)))))))

C

float code(float u, float s) {
	return s * -logf((1.0f + (-4.0f * (((0.25f * (u * ((float) M_PI))) - (((float) M_PI) * 0.25f)) / s))));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(1.0) + Float32(Float32(-4.0) * Float32(Float32(Float32(Float32(0.25) * Float32(u * Float32(pi))) - Float32(Float32(pi) * Float32(0.25))) / s))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(1.0) + (single(-4.0) * (((single(0.25) * (u * single(pi))) - (single(pi) * single(0.25))) / s))));
end

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 24.8%

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

5. Taylor expanded in u around 0 25.0%

   \[\leadsto s \cdot \left(-\log \left(-4 \cdot \frac{0.25 \cdot \left(u \cdot \pi\right) - \color{blue}{0.25 \cdot \pi}}{s} + 1\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 25.0%

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

7. Simplified 25.0%

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

8. Final simplification 25.0%

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

Alternative 4: 25.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(1 + -4 \cdot \left(\frac{\pi}{s} \cdot -0.25\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (- (log (+ 1.0 (* -4.0 (* (/ PI s) -0.25)))))))

C

float code(float u, float s) {
	return s * -logf((1.0f + (-4.0f * ((((float) M_PI) / s) * -0.25f))));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(1.0) + Float32(Float32(-4.0) * Float32(Float32(Float32(pi) / s) * Float32(-0.25)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(1.0) + (single(-4.0) * ((single(pi) / s) * single(-0.25)))));
end

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 24.8%

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

5. Taylor expanded in u around 0 25.0%

   \[\leadsto s \cdot \left(-\log \left(-4 \cdot \color{blue}{\left(-0.25 \cdot \frac{\pi}{s}\right)} + 1\right)\right) \]
6. Step-by-step derivation

   1. *-commutative 25.0%

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

7. Simplified 25.0%

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

8. Final simplification 25.0%

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

Alternative 5: 11.6% accurate, 6.8× 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 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 11.6%

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

   1. associate--r+ 11.6%

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

   2. cancel-sign-sub-inv 11.6%

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

   3. distribute-rgt-out-- 11.6%

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

   4. *-commutative 11.6%

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

   5. metadata-eval 11.6%

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

   6. metadata-eval 11.6%

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

   7. *-commutative 11.6%

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

6. Simplified 11.6%

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

7. Taylor expanded in u around 0 11.6%

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

   1. +-commutative 11.6%

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

   2. associate-*r* 11.6%

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

   3. *-commutative 11.6%

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

   4. distribute-rgt-out 11.6%

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

9. Simplified 11.6%

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

10. Final simplification 11.6%

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

Alternative 6: 11.4% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. distribute-lft-neg-out 99.1%

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

   2. distribute-rgt-neg-in 99.1%

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

   3. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in u around 0 11.2%

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

   1. mul-1-neg 11.2%

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

6. Simplified 11.2%

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

7. Final simplification 11.2%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023240 
(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))))