Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 14.4s

Alternatives: 12

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.3× speedup

Math

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

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 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. Add Preprocessing

Alternative 2: 86.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ PI (- s)))))
      (/ (- 1.0 u) (+ 1.0 (+ 1.0 (/ 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 + (1.0f + (((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) + Float32(Float32(1.0) + 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) + (single(1.0) + (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 s around inf 87.1%

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

6. Final simplification 87.1%

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

Alternative 3: 25.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\pi}{s}\\ -4 \cdot \frac{s \cdot \left(u \cdot \left(\frac{\pi}{s} \cdot -0.25 - \frac{\pi}{s} \cdot 0.25\right)\right)}{t\_0} - s \cdot \log t\_0 \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ PI s))))
   (-
    (* -4.0 (/ (* s (* u (- (* (/ PI s) -0.25) (* (/ PI s) 0.25)))) t_0))
    (* s (log t_0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f + (((float) M_PI) / s);
	return (-4.0f * ((s * (u * (((((float) M_PI) / s) * -0.25f) - ((((float) M_PI) / s) * 0.25f)))) / t_0)) - (s * logf(t_0));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + Float32(Float32(pi) / s))
	return Float32(Float32(Float32(-4.0) * Float32(Float32(s * Float32(u * Float32(Float32(Float32(Float32(pi) / s) * Float32(-0.25)) - Float32(Float32(Float32(pi) / s) * Float32(0.25))))) / t_0)) - Float32(s * log(t_0)))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + (single(pi) / s);
	tmp = (single(-4.0) * ((s * (u * (((single(pi) / s) * single(-0.25)) - ((single(pi) / s) * single(0.25))))) / t_0)) - (s * log(t_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 s around -inf 24.8%

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

6. Taylor expanded in u around 0 25.1%

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

7. Final simplification 25.1%

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

Alternative 4: 25.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(u \cdot \frac{\pi}{1 + \frac{\pi}{s}}\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (- (* 2.0 (* u (/ PI (+ 1.0 (/ PI s))))) (* s (log1p (/ PI s)))))

C

float code(float u, float s) {
	return (2.0f * (u * (((float) M_PI) / (1.0f + (((float) M_PI) / s))))) - (s * log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(Float32(2.0) * Float32(u * Float32(Float32(pi) / Float32(Float32(1.0) + Float32(Float32(pi) / s))))) - 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 24.8%

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

   1. add-log-exp 14.2%

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

   2. *-commutative 14.2%

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

   3. exp-to-pow 14.2%

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

7. Applied egg-rr 14.2%

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

9. Simplified 14.2%

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

10. Taylor expanded in u around 0 25.1%

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

    1. +-commutative 25.1%

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

    2. mul-1-neg 25.1%

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

    3. unsub-neg 25.1%

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

    4. associate-/l* 25.1%

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

    5. log1p-define 25.1%

       \[\leadsto 2 \cdot \left(u \cdot \frac{\pi}{1 + \frac{\pi}{s}}\right) - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

12. Simplified 25.1%

    \[\leadsto \color{blue}{2 \cdot \left(u \cdot \frac{\pi}{1 + \frac{\pi}{s}}\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]
13. Add Preprocessing

Alternative 5: 25.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(Float32(u * Float32(-Float32(-2.0))) - 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 24.8%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} + 4 \cdot \frac{u \cdot \left(-0.25 \cdot \frac{\pi}{s} - 0.25 \cdot \frac{\pi}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

   2. associate-*r/ 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \color{blue}{\frac{4 \cdot \left(u \cdot \left(-0.25 \cdot \frac{\pi}{s} - 0.25 \cdot \frac{\pi}{s}\right)\right)}{1 + \frac{\pi}{s}}}\right) \]

   3. associate-*r/ 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \left(\color{blue}{\frac{-0.25 \cdot \pi}{s}} - 0.25 \cdot \frac{\pi}{s}\right)\right)}{1 + \frac{\pi}{s}}\right) \]

   4. associate-*r/ 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \left(\frac{-0.25 \cdot \pi}{s} - \color{blue}{\frac{0.25 \cdot \pi}{s}}\right)\right)}{1 + \frac{\pi}{s}}\right) \]

   5. div-sub 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \color{blue}{\frac{-0.25 \cdot \pi - 0.25 \cdot \pi}{s}}\right)}{1 + \frac{\pi}{s}}\right) \]

   6. associate-/l* 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \color{blue}{\frac{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right)}{s}}}{1 + \frac{\pi}{s}}\right) \]

   7. distribute-rgt-out-- 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \frac{u \cdot \color{blue}{\left(\pi \cdot \left(-0.25 - 0.25\right)\right)}}{s}}{1 + \frac{\pi}{s}}\right) \]

   8. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right)}{s}}{1 + \frac{\pi}{s}}\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right)}{s}}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0 25.1%

   \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \color{blue}{-2 \cdot u}\right) \]
10. Step-by-step derivation

    1. *-commutative 25.1%

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

11. Simplified 25.1%

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

12. Final simplification 25.1%

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

Alternative 6: 25.1% accurate, 4.0× speedup

Math

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

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(Float32(-s) * 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 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 24.8%

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

6. Taylor expanded in u around 0 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\log \left(1 + \frac{\pi}{s}\right)} \]
7. 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 24.8%

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

6. Taylor expanded in u around 0 25.1%

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

   1. associate-*r* 25.1%

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

   2. neg-mul-1 25.1%

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

   3. log1p-define 25.1%

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

8. Simplified 25.1%

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

Alternative 8: 11.4% accurate, 30.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * ((single(pi) / -s) - (single(-2.0) * ((u * 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 s around -inf 10.8%

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

   1. associate-*r/ 10.8%

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

   2. associate--r+ 10.8%

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

   3. cancel-sign-sub-inv 10.8%

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

   4. cancel-sign-sub-inv 10.8%

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

   5. metadata-eval 10.8%

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

   6. associate-*r* 10.8%

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

   7. distribute-rgt-out 10.8%

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

   8. metadata-eval 10.8%

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

   9. *-commutative 10.8%

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

   10. associate-*r* 10.8%

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

   11. *-commutative 10.8%

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

7. Simplified 10.8%

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

8. Taylor expanded in u around 0 10.8%

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

9. Final simplification 10.8%

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

Alternative 9: 11.5% 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 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 10.8%

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

   1. associate--r+ 10.8%

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

   2. cancel-sign-sub-inv 10.8%

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

   3. cancel-sign-sub-inv 10.8%

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

   4. metadata-eval 10.8%

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

   5. associate-*r* 10.8%

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

   6. distribute-rgt-out 10.8%

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

   7. metadata-eval 10.8%

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

   8. *-commutative 10.8%

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

   9. associate-*r* 10.8%

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

   10. *-commutative 10.8%

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

7. Simplified 10.8%

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

8. Taylor expanded in u around 0 10.8%

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

   1. neg-mul-1 10.8%

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

   2. +-commutative 10.8%

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

   3. associate-*r* 10.8%

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

   4. neg-mul-1 10.8%

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

   5. distribute-rgt-out 10.8%

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

10. Simplified 10.8%

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

11. Final simplification 10.8%

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

Alternative 10: 11.3% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (s * -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 10.6%

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

   1. associate-*r/ 10.6%

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

7. Applied egg-rr 10.6%

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

8. Final simplification 10.6%

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

Alternative 11: 11.2% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * (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 10.6%

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

Alternative 12: 11.3% 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 10.6%

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

   1. neg-mul-1 10.6%

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

7. Simplified 10.6%

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

Reproduce

herbie shell --seed 2024169 
(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))))