Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 15.0s

Alternatives: 10

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 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

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 3: 25.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 25.1%

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

   1. cancel-sign-sub-inv 25.1%

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

   2. distribute-rgt-out-- 25.1%

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

   3. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + \left(--0.25\right) \cdot \pi}{s}\right) \]

   4. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{0.25} \cdot \pi}{s}\right) \]

   5. *-commutative 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]

6. Taylor expanded in u around 0 25.3%

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

7. Taylor expanded in s around 0 25.3%

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

8. Taylor expanded in s around 0 25.4%

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

   1. mul-1-neg 25.4%

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

   2. unsub-neg 25.4%

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

10. Simplified 25.4%

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

11. Final simplification 25.4%

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

Alternative 4: 25.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(\left(--2\right) - \frac{\mathsf{log1p}\left(\frac{\pi}{s}\right)}{u}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (* u (- (- -2.0) (/ (log1p (/ PI s)) u)))))

C

float code(float u, float s) {
	return s * (u * (-(-2.0f) - (log1pf((((float) M_PI) / s)) / u)));
}

Julia

function code(u, s)
	return Float32(s * Float32(u * Float32(Float32(-Float32(-2.0)) - Float32(log1p(Float32(Float32(pi) / s)) / u))))
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 25.1%

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

   1. cancel-sign-sub-inv 25.1%

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

   2. distribute-rgt-out-- 25.1%

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

   3. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + \left(--0.25\right) \cdot \pi}{s}\right) \]

   4. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{0.25} \cdot \pi}{s}\right) \]

   5. *-commutative 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]

6. Taylor expanded in u around 0 25.3%

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

7. Taylor expanded in s around 0 25.3%

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

8. Taylor expanded in u around inf 25.3%

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

   1. sub-neg 25.3%

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

   2. log1p-define 25.3%

      \[\leadsto \left(-s\right) \cdot \left(u \cdot \left(\frac{\color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)}}{u} + \left(-2\right)\right)\right) \]

   3. metadata-eval 25.3%

      \[\leadsto \left(-s\right) \cdot \left(u \cdot \left(\frac{\mathsf{log1p}\left(\frac{\pi}{s}\right)}{u} + \color{blue}{-2}\right)\right) \]

10. Simplified 25.3%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\left(u \cdot \left(\frac{\mathsf{log1p}\left(\frac{\pi}{s}\right)}{u} + -2\right)\right)} \]

11. Final simplification 25.3%

    \[\leadsto s \cdot \left(u \cdot \left(\left(--2\right) - \frac{\mathsf{log1p}\left(\frac{\pi}{s}\right)}{u}\right)\right) \]
12. Add Preprocessing

Alternative 5: 25.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 25.1%

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

   1. cancel-sign-sub-inv 25.1%

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

   2. distribute-rgt-out-- 25.1%

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

   3. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + \left(--0.25\right) \cdot \pi}{s}\right) \]

   4. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{0.25} \cdot \pi}{s}\right) \]

   5. *-commutative 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]

6. Taylor expanded in u around 0 25.3%

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

7. Taylor expanded in s around 0 25.3%

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

8. Final simplification 25.3%

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

Alternative 6: 25.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(1.0) + (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. Add Preprocessing

3. Taylor expanded in s around -inf 25.1%

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

   1. cancel-sign-sub-inv 25.1%

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

   2. distribute-rgt-out-- 25.1%

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

   3. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + \left(--0.25\right) \cdot \pi}{s}\right) \]

   4. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{0.25} \cdot \pi}{s}\right) \]

   5. *-commutative 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]

6. Taylor expanded in u around 0 25.3%

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

7. Final simplification 25.3%

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

3. Taylor expanded in s around -inf 25.1%

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

   1. cancel-sign-sub-inv 25.1%

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

   2. distribute-rgt-out-- 25.1%

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

   3. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + \left(--0.25\right) \cdot \pi}{s}\right) \]

   4. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{0.25} \cdot \pi}{s}\right) \]

   5. *-commutative 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]

6. Taylor expanded in u around 0 25.3%

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

   1. log1p-define 25.3%

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

8. Simplified 25.3%

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

9. Final simplification 25.3%

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

Alternative 8: 11.6% accurate, 54.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -single(pi) - (single(-2.0) * (u * 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. Add Preprocessing

3. Taylor expanded in s around -inf 25.1%

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

   1. cancel-sign-sub-inv 25.1%

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

   2. distribute-rgt-out-- 25.1%

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

   3. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + \left(--0.25\right) \cdot \pi}{s}\right) \]

   4. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{0.25} \cdot \pi}{s}\right) \]

   5. *-commutative 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]

6. Taylor expanded in u around 0 25.3%

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

7. Taylor expanded in s around -inf 9.0%

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

   1. mul-1-neg 9.0%

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

   2. distribute-neg-frac2 9.0%

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

   3. +-commutative 9.0%

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

   4. associate-/l* 9.0%

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

   5. mul-1-neg 9.0%

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

   6. distribute-rgt-neg-out 9.0%

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

   7. distribute-lft-out 9.0%

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

9. Simplified 9.0%

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

10. Taylor expanded in s around inf 12.1%

    \[\leadsto \color{blue}{-1 \cdot \left(\pi + -2 \cdot \left(u \cdot \pi\right)\right)} \]
11. Step-by-step derivation

    1. mul-1-neg 12.1%

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

    2. *-commutative 12.1%

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

12. Simplified 12.1%

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

13. Final simplification 12.1%

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

Alternative 9: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(pi) * Float32(Float32(-1.0) + Float32(u * Float32(2.0))))
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 25.1%

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

   1. cancel-sign-sub-inv 25.1%

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

   2. distribute-rgt-out-- 25.1%

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

   3. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + \left(--0.25\right) \cdot \pi}{s}\right) \]

   4. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{0.25} \cdot \pi}{s}\right) \]

   5. *-commutative 25.1%

      \[\leadsto \left(-s\right) \cdot \log \left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) \]

5. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(1 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} \]

6. Taylor expanded in u around 0 25.3%

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

7. Taylor expanded in s around -inf 9.0%

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

   1. mul-1-neg 9.0%

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

   2. distribute-neg-frac2 9.0%

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

   3. +-commutative 9.0%

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

   4. associate-/l* 9.0%

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

   5. mul-1-neg 9.0%

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

   6. distribute-rgt-neg-out 9.0%

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

   7. distribute-lft-out 9.0%

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

9. Simplified 9.0%

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

10. Taylor expanded in s around inf 12.1%

    \[\leadsto \color{blue}{-1 \cdot \left(\pi + -2 \cdot \left(u \cdot \pi\right)\right)} \]
11. Step-by-step derivation

    1. distribute-lft-in 12.1%

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

    2. associate-*r* 12.1%

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

    3. metadata-eval 12.1%

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

    4. +-commutative 12.1%

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

    5. associate-*r* 12.1%

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

    6. distribute-rgt-out 12.1%

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

    7. *-commutative 12.1%

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

12. Simplified 12.1%

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

13. Final simplification 12.1%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]
14. Add Preprocessing

Alternative 10: 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 99.1%

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

3. Simplified 99.0%

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

5. Taylor expanded in u around 0 11.9%

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

   1. mul-1-neg 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 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))))