Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 13.4s

Alternatives: 9

Speedup: 1.0×

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

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

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 (exp (/ PI (- s))))) (/ u (- -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 + expf((((float) M_PI) / -s)))) + (u / (-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(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(u / 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) + exp((single(pi) / -s)))) + (u / (single(-1.0) - t_0))))) + single(-1.0)));
end

Derivation

1. Initial program 99.1%

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

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 s around 0 99.0%

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

   1. mul-1-neg 99.0%

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

   2. sub-neg 99.0%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

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%

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

Alternative 3: 25.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(-1 + \left(2 + 4 \cdot \frac{\pi \cdot 0.25}{s}\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (- (log (+ -1.0 (+ 2.0 (* 4.0 (/ (* PI 0.25) s))))))))

C

float code(float u, float s) {
	return s * -logf((-1.0f + (2.0f + (4.0f * ((((float) M_PI) * 0.25f) / s)))));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(-1.0) + Float32(Float32(2.0) + Float32(Float32(4.0) * Float32(Float32(Float32(pi) * Float32(0.25)) / s)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(-1.0) + (single(2.0) + (single(4.0) * ((single(pi) * single(0.25)) / s)))));
end

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around -inf 24.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 + 4 \cdot \frac{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right) - -0.25 \cdot \pi}{s}\right)} - 1\right) \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv 24.4%

      \[\leadsto \left(-s\right) \cdot \log \left(\left(2 + 4 \cdot \frac{\color{blue}{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right) + \left(--0.25\right) \cdot \pi}}{s}\right) - 1\right) \]

   2. metadata-eval 24.4%

      \[\leadsto \left(-s\right) \cdot \log \left(\left(2 + 4 \cdot \frac{u \cdot \left(-0.25 \cdot \pi - 0.25 \cdot \pi\right) + \color{blue}{0.25} \cdot \pi}{s}\right) - 1\right) \]

   3. distribute-rgt-out-- 24.4%

      \[\leadsto \left(-s\right) \cdot \log \left(\left(2 + 4 \cdot \frac{u \cdot \color{blue}{\left(\pi \cdot \left(-0.25 - 0.25\right)\right)} + 0.25 \cdot \pi}{s}\right) - 1\right) \]

   4. metadata-eval 24.4%

      \[\leadsto \left(-s\right) \cdot \log \left(\left(2 + 4 \cdot \frac{u \cdot \left(\pi \cdot \color{blue}{-0.5}\right) + 0.25 \cdot \pi}{s}\right) - 1\right) \]

   5. *-commutative 24.4%

      \[\leadsto \left(-s\right) \cdot \log \left(\left(2 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \color{blue}{\pi \cdot 0.25}}{s}\right) - 1\right) \]

5. Simplified 24.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\left(2 + 4 \cdot \frac{u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25}{s}\right)} - 1\right) \]

6. Taylor expanded in u around 0 24.6%

   \[\leadsto \left(-s\right) \cdot \log \left(\left(2 + 4 \cdot \frac{\color{blue}{0.25 \cdot \pi}}{s}\right) - 1\right) \]

7. Final simplification 24.6%

   \[\leadsto s \cdot \left(-\log \left(-1 + \left(2 + 4 \cdot \frac{\pi \cdot 0.25}{s}\right)\right)\right) \]
8. Add Preprocessing

Alternative 4: 14.3% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 2.9999999047965676 \cdot 10^{-20}:\\ \;\;\;\;s \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\pi \cdot \left(u \cdot \left(-0.5 + \frac{0.25}{u}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (if (<= s 2.9999999047965676e-20)
   (* s 0.0)
   (* -4.0 (* PI (* u (+ -0.5 (/ 0.25 u)))))))

C

float code(float u, float s) {
	float tmp;
	if (s <= 2.9999999047965676e-20f) {
		tmp = s * 0.0f;
	} else {
		tmp = -4.0f * (((float) M_PI) * (u * (-0.5f + (0.25f / u))));
	}
	return tmp;
}

Julia

function code(u, s)
	tmp = Float32(0.0)
	if (s <= Float32(2.9999999047965676e-20))
		tmp = Float32(s * Float32(0.0));
	else
		tmp = Float32(Float32(-4.0) * Float32(Float32(pi) * Float32(u * Float32(Float32(-0.5) + Float32(Float32(0.25) / u)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(u, s)
	tmp = single(0.0);
	if (s <= single(2.9999999047965676e-20))
		tmp = s * single(0.0);
	else
		tmp = single(-4.0) * (single(pi) * (u * (single(-0.5) + (single(0.25) / u))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 2.9999999e-20

   1. Initial program 99.3%

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

   3. Taylor expanded in s around inf 13.5%

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

   if 2.9999999e-20 < s

   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 15.8%

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

      1. associate--r+ 15.8%

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

      2. cancel-sign-sub-inv 15.8%

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

      3. metadata-eval 15.8%

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

      4. cancel-sign-sub-inv 15.8%

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

      5. associate-*r* 15.8%

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

      6. distribute-rgt-out 15.8%

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

      7. metadata-eval 15.8%

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

      8. *-commutative 15.8%

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

      9. *-commutative 15.8%

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

   7. Simplified 15.8%

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

      1. add-cube-cbrt 15.9%

         \[\leadsto -4 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

      2. pow3 15.9%

         \[\leadsto -4 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

   9. Applied egg-rr 15.9%

      \[\leadsto -4 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

   10. Taylor expanded in u around inf 15.8%

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

       1. *-commutative 15.8%

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

       2. fma-define 15.8%

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

       3. associate-*r/ 15.8%

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

       4. *-commutative 15.8%

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

       5. associate-/l* 15.8%

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

   12. Simplified 15.8%

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

   13. Taylor expanded in u around inf 15.8%

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

       1. distribute-rgt-in 15.8%

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

       2. *-commutative 15.8%

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

       3. associate-*r/ 15.8%

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

       4. *-commutative 15.8%

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

       5. associate-*r/ 15.8%

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

       6. distribute-rgt-in 15.8%

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

       7. distribute-lft-in 15.8%

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

       8. *-commutative 15.8%

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

       9. associate-*l* 15.8%

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

   15. Simplified 15.8%

       \[\leadsto -4 \cdot \color{blue}{\left(\pi \cdot \left(\left(-0.5 + \frac{0.25}{u}\right) \cdot u\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.9999999047965676 \cdot 10^{-20}:\\ \;\;\;\;s \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\pi \cdot \left(u \cdot \left(-0.5 + \frac{0.25}{u}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 11.6% accurate, 39.4× speedup

Math

\[\begin{array}{l} \\ -4 \cdot \left(\pi \cdot \left(u \cdot \left(-0.5 + \frac{0.25}{u}\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (* -4.0 (* PI (* u (+ -0.5 (/ 0.25 u))))))

C

float code(float u, float s) {
	return -4.0f * (((float) M_PI) * (u * (-0.5f + (0.25f / u))));
}

Julia

function code(u, s)
	return Float32(Float32(-4.0) * Float32(Float32(pi) * Float32(u * Float32(Float32(-0.5) + Float32(Float32(0.25) / u)))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(-4.0) * (single(pi) * (u * (single(-0.5) + (single(0.25) / u))));
end

Derivation

1. Initial program 99.1%

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

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 s around -inf 11.3%

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

   1. associate--r+ 11.3%

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

   2. cancel-sign-sub-inv 11.3%

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

   3. metadata-eval 11.3%

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

   4. cancel-sign-sub-inv 11.3%

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

   5. associate-*r* 11.3%

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

   6. distribute-rgt-out 11.3%

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

   7. metadata-eval 11.3%

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

   8. *-commutative 11.3%

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

   9. *-commutative 11.3%

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

7. Simplified 11.3%

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

   1. add-cube-cbrt 11.3%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

   2. pow3 11.3%

      \[\leadsto -4 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

9. Applied egg-rr 11.3%

   \[\leadsto -4 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

10. Taylor expanded in u around inf 11.3%

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

    1. *-commutative 11.3%

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

    2. fma-define 11.3%

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

    3. associate-*r/ 11.3%

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

    4. *-commutative 11.3%

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

    5. associate-/l* 11.3%

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

12. Simplified 11.3%

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

13. Taylor expanded in u around inf 11.3%

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

    1. distribute-rgt-in 11.3%

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

    2. *-commutative 11.3%

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

    3. associate-*r/ 11.3%

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

    4. *-commutative 11.3%

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

    5. associate-*r/ 11.3%

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

    6. distribute-rgt-in 11.3%

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

    7. distribute-lft-in 11.3%

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

    8. *-commutative 11.3%

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

    9. associate-*l* 11.3%

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

15. Simplified 11.3%

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

16. Final simplification 11.3%

    \[\leadsto -4 \cdot \left(\pi \cdot \left(u \cdot \left(-0.5 + \frac{0.25}{u}\right)\right)\right) \]
17. Add Preprocessing

Alternative 6: 11.6% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

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 s around -inf 11.3%

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

   1. associate--r+ 11.3%

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

   2. cancel-sign-sub-inv 11.3%

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

   3. metadata-eval 11.3%

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

   4. cancel-sign-sub-inv 11.3%

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

   5. associate-*r* 11.3%

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

   6. distribute-rgt-out 11.3%

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

   7. metadata-eval 11.3%

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

   8. *-commutative 11.3%

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

   9. *-commutative 11.3%

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

7. Simplified 11.3%

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

   1. add-cube-cbrt 11.3%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

   2. pow3 11.3%

      \[\leadsto -4 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

9. Applied egg-rr 11.3%

   \[\leadsto -4 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

10. Taylor expanded in u around 0 11.3%

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

11. Taylor expanded in u around inf 11.3%

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

    1. +-commutative 11.3%

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

    2. mul-1-neg 11.3%

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

    3. unsub-neg 11.3%

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

    4. *-commutative 11.3%

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

13. Simplified 11.3%

    \[\leadsto \color{blue}{u \cdot \left(\pi \cdot 2 - \frac{\pi}{u}\right)} \]
14. Add Preprocessing

Alternative 7: 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%

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

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 s around -inf 11.3%

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

   1. associate--r+ 11.3%

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

   2. cancel-sign-sub-inv 11.3%

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

   3. metadata-eval 11.3%

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

   4. cancel-sign-sub-inv 11.3%

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

   5. associate-*r* 11.3%

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

   6. distribute-rgt-out 11.3%

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

   7. metadata-eval 11.3%

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

   8. *-commutative 11.3%

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

   9. *-commutative 11.3%

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

7. Simplified 11.3%

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

   1. add-cube-cbrt 11.3%

      \[\leadsto -4 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

   2. pow3 11.3%

      \[\leadsto -4 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

9. Applied egg-rr 11.3%

   \[\leadsto -4 \cdot \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot \left(-0.25 \cdot u + 0.25\right) + \left(\pi \cdot u\right) \cdot -0.25\right) \]

10. Taylor expanded in u around 0 11.3%

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

    1. associate-*r* 11.3%

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

    2. distribute-rgt-out 11.3%

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

12. Simplified 11.3%

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

13. Final simplification 11.3%

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

Alternative 8: 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 11.1%

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

   1. mul-1-neg 11.1%

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

7. Simplified 11.1%

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

Alternative 9: 4.6% accurate, 433.0× 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(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. Add Preprocessing

3. Taylor expanded in u around 0 11.1%

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

   1. associate-*r/ 11.1%

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

   2. add-sqr-sqrt -0.0%

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

   3. sqrt-unprod 8.9%

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

   4. sqr-neg 8.9%

      \[\leadsto \frac{\sqrt{\color{blue}{s \cdot s}} \cdot \pi}{s} \]

   5. sqrt-unprod 4.7%

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

   6. add-sqr-sqrt 4.7%

      \[\leadsto \frac{\color{blue}{s} \cdot \pi}{s} \]

5. Applied egg-rr 4.7%

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

   1. associate-/l* 4.7%

      \[\leadsto \color{blue}{s \cdot \frac{\pi}{s}} \]

7. Simplified 4.7%

   \[\leadsto \color{blue}{s \cdot \frac{\pi}{s}} \]

8. Taylor expanded in s around 0 4.7%

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

Reproduce

herbie shell --seed 2024111 
(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))))