Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 19.5s

Alternatives: 7

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (+
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ (- PI) s))))
      (* (- 1.0 u) (/ 1.0 (+ 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 / (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(-Float32(pi)) / s)))) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) / 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) / (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. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 3: 25.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * abs(log((s * 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. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around -inf 25.1%

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

5. Taylor expanded in s around 0 25.1%

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

6. Taylor expanded in u around 0 25.3%

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

   1. +-commutative 25.3%

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

   2. mul-1-neg 25.3%

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

8. Simplified 25.3%

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

   1. add-sqr-sqrt 25.3%

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

   2. sqrt-unprod 25.3%

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

   3. pow1 25.3%

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

   4. pow1 25.3%

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

   5. pow-sqr 25.3%

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

   6. +-commutative 25.3%

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

   7. add-sqr-sqrt 25.3%

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

   8. sqrt-unprod 25.3%

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

   9. sqr-neg 25.3%

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

   10. sqrt-unprod -0.0%

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

   11. add-sqr-sqrt 25.4%

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

   12. metadata-eval 25.4%

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

10. Applied egg-rr 25.4%

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

    1. unpow2 25.4%

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

    2. rem-sqrt-square 25.4%

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

    3. log-prod 25.4%

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

    4. *-commutative 25.4%

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

12. Simplified 25.4%

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

13. Final simplification 25.4%

    \[\leadsto \left(-s\right) \cdot \left|\log \left(s \cdot \pi\right)\right| \]

Alternative 4: 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 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. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around -inf 25.1%

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

5. Taylor expanded in u around 0 25.2%

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

   1. mul-1-neg 25.2%

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

   2. log1p-def 25.2%

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

7. Simplified 25.2%

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

8. Final simplification 25.2%

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

Alternative 5: 12.2% accurate, 54.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (-single(pi) * (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. sub-neg 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 11.4%

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

5. Taylor expanded in u around inf 5.0%

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

   1. distribute-rgt-out-- 5.0%

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

   2. metadata-eval 5.0%

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

   3. *-commutative 5.0%

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

   4. associate-*l* 5.0%

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

   5. *-commutative 5.0%

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

7. Simplified 5.0%

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

   1. add-sqr-sqrt 5.0%

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

   2. sqrt-unprod 5.0%

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

   3. sqr-neg 5.0%

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

   4. sqrt-unprod -0.0%

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

   5. add-sqr-sqrt 12.2%

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

   6. neg-sub0 12.2%

      \[\leadsto 4 \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot \left(u \cdot 0.5\right)\right) \]

9. Applied egg-rr 12.2%

   \[\leadsto 4 \cdot \left(\color{blue}{\left(0 - \pi\right)} \cdot \left(u \cdot 0.5\right)\right) \]

10. Final simplification 12.2%

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

Alternative 6: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (u * (single(pi) * single(2.0))) - 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. Taylor expanded in s around -inf 25.1%

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

   1. associate-*r/ 25.1%

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

   2. associate-/l* 25.1%

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

   3. cancel-sign-sub-inv 25.1%

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

   4. metadata-eval 25.1%

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

   5. +-commutative 25.1%

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

   6. *-commutative 25.1%

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

   7. distribute-rgt-out-- 25.1%

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

   8. metadata-eval 25.1%

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

4. Simplified 25.1%

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

5. Taylor expanded in s around inf 11.4%

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

   1. associate-*r* 11.4%

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

   2. *-commutative 11.4%

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

   3. *-commutative 11.4%

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

   4. distribute-lft-in 11.4%

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

   5. +-commutative 11.4%

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

   6. associate-*r* 11.4%

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

   7. metadata-eval 11.4%

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

   8. neg-mul-1 11.4%

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

   9. associate-*r* 11.4%

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

   10. *-commutative 11.4%

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

   11. associate-*r* 11.4%

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

   12. metadata-eval 11.4%

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

   13. associate-*r* 11.4%

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

   14. *-commutative 11.4%

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

7. Simplified 11.4%

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

8. Final simplification 11.4%

   \[\leadsto u \cdot \left(\pi \cdot 2\right) - \pi \]

Alternative 7: 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 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. sub-neg 99.1%

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

3. Simplified 99.1%

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

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 2023336 
(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))))