Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 14.9s

Alternatives: 5

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(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 98.9%

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

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%

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

Alternative 2: 25.2% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * (s * u)) - (s * log((single(1.0) + (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. Taylor expanded in s around -inf 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

7. Simplified 24.7%

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

8. Taylor expanded in u around 0 25.0%

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

9. Taylor expanded in s around 0 25.0%

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

    1. *-commutative 25.0%

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

11. Simplified 25.0%

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

12. Final simplification 25.0%

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

Alternative 3: 25.2% accurate, 3.6× 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 98.9%

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

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 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

7. Simplified 24.7%

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

8. Taylor expanded in u around 0 25.0%

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

   1. associate-*r* 25.0%

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

   2. neg-mul-1 25.0%

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

   3. log1p-def 25.0%

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

10. Simplified 25.0%

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

11. Final simplification 25.0%

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

Alternative 4: 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 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. div-inv 98.8%

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

   2. add-cube-cbrt 98.9%

      \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{1}{s}}}} + -1\right)\right) \]

   3. associate-*l* 98.8%

      \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\sqrt[3]{\pi} \cdot \frac{1}{s}\right)}}}} + -1\right)\right) \]

   4. pow2 98.8%

      \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \left(\sqrt[3]{\pi} \cdot \frac{1}{s}\right)}}} + -1\right)\right) \]

6. Applied egg-rr 98.8%

   \[\leadsto s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot \frac{1}{s}\right)}}}} + -1\right)\right) \]

7. Taylor expanded in s around inf 11.3%

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

   1. associate--r+ 11.3%

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

   2. cancel-sign-sub-inv 11.3%

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

   3. *-commutative 11.3%

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

   4. *-commutative 11.3%

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

   5. associate-*r* 11.3%

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

   6. *-commutative 11.3%

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

   7. *-commutative 11.3%

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

   8. associate-*r* 11.3%

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

   9. distribute-lft-out-- 11.3%

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

   10. metadata-eval 11.3%

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

   11. *-commutative 11.3%

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

   12. *-commutative 11.3%

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

   13. metadata-eval 11.3%

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

   14. +-commutative 11.3%

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

   15. associate-*r* 11.3%

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

   16. *-commutative 11.3%

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

9. Simplified 11.3%

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

10. Final simplification 11.3%

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

Alternative 5: 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 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 11.1%

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

   1. neg-mul-1 11.1%

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

7. Simplified 11.1%

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

8. Final simplification 11.1%

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

Reproduce

herbie shell --seed 2024011 
(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))))