Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 9.9s

Alternatives: 7

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: 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(-Float32(pi)) / 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%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 2: 11.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 4 \cdot {\left(\sqrt[3]{\pi \cdot -0.25 + 0.5 \cdot \left(u \cdot \pi\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* 4.0 (pow (cbrt (+ (* PI -0.25) (* 0.5 (* u PI)))) 3.0)))

C

float code(float u, float s) {
	return 4.0f * powf(cbrtf(((((float) M_PI) * -0.25f) + (0.5f * (u * ((float) M_PI))))), 3.0f);
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * (cbrt(Float32(Float32(Float32(pi) * Float32(-0.25)) + Float32(Float32(0.5) * Float32(u * Float32(pi))))) ^ Float32(3.0)))
end

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. 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. Step-by-step derivation

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. distribute-rgt-out-- 11.4%

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

   4. *-commutative 11.4%

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

   5. metadata-eval 11.4%

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

   6. metadata-eval 11.4%

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

   7. *-commutative 11.4%

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

6. Simplified 11.4%

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

   1. add-cube-cbrt 11.4%

      \[\leadsto 4 \cdot \color{blue}{\left(\left(\sqrt[3]{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25} \cdot \sqrt[3]{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}\right) \cdot \sqrt[3]{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}\right)} \]

   2. pow3 11.4%

      \[\leadsto 4 \cdot \color{blue}{{\left(\sqrt[3]{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}\right)}^{3}} \]

   3. +-commutative 11.4%

      \[\leadsto 4 \cdot {\left(\sqrt[3]{\color{blue}{\pi \cdot -0.25 + \left(\pi \cdot u\right) \cdot 0.5}}\right)}^{3} \]

   4. fma-def 11.4%

      \[\leadsto 4 \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(\pi, -0.25, \left(\pi \cdot u\right) \cdot 0.5\right)}}\right)}^{3} \]

   5. associate-*l* 11.4%

      \[\leadsto 4 \cdot {\left(\sqrt[3]{\mathsf{fma}\left(\pi, -0.25, \color{blue}{\pi \cdot \left(u \cdot 0.5\right)}\right)}\right)}^{3} \]

8. Applied egg-rr 11.4%

   \[\leadsto 4 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\pi, -0.25, \pi \cdot \left(u \cdot 0.5\right)\right)}\right)}^{3}} \]

9. Taylor expanded in u around 0 11.4%

   \[\leadsto 4 \cdot {\left(\sqrt[3]{\color{blue}{-0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)}}\right)}^{3} \]

10. Final simplification 11.4%

    \[\leadsto 4 \cdot {\left(\sqrt[3]{\pi \cdot -0.25 + 0.5 \cdot \left(u \cdot \pi\right)}\right)}^{3} \]

Alternative 3: 11.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ 4 \cdot {\left(\sqrt[3]{\pi \cdot \left(-0.25 + u \cdot 0.5\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* 4.0 (pow (cbrt (* PI (+ -0.25 (* u 0.5)))) 3.0)))

C

float code(float u, float s) {
	return 4.0f * powf(cbrtf((((float) M_PI) * (-0.25f + (u * 0.5f)))), 3.0f);
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * (cbrt(Float32(Float32(pi) * Float32(Float32(-0.25) + Float32(u * Float32(0.5))))) ^ Float32(3.0)))
end

Derivation

1. Initial program 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. 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. Step-by-step derivation

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. distribute-rgt-out-- 11.4%

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

   4. *-commutative 11.4%

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

   5. metadata-eval 11.4%

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

   6. metadata-eval 11.4%

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

   7. *-commutative 11.4%

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

6. Simplified 11.4%

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

   1. add-cube-cbrt 11.4%

      \[\leadsto 4 \cdot \color{blue}{\left(\left(\sqrt[3]{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25} \cdot \sqrt[3]{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}\right) \cdot \sqrt[3]{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}\right)} \]

   2. pow3 11.4%

      \[\leadsto 4 \cdot \color{blue}{{\left(\sqrt[3]{\left(\pi \cdot u\right) \cdot 0.5 + \pi \cdot -0.25}\right)}^{3}} \]

   3. +-commutative 11.4%

      \[\leadsto 4 \cdot {\left(\sqrt[3]{\color{blue}{\pi \cdot -0.25 + \left(\pi \cdot u\right) \cdot 0.5}}\right)}^{3} \]

   4. fma-def 11.4%

      \[\leadsto 4 \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(\pi, -0.25, \left(\pi \cdot u\right) \cdot 0.5\right)}}\right)}^{3} \]

   5. associate-*l* 11.4%

      \[\leadsto 4 \cdot {\left(\sqrt[3]{\mathsf{fma}\left(\pi, -0.25, \color{blue}{\pi \cdot \left(u \cdot 0.5\right)}\right)}\right)}^{3} \]

8. Applied egg-rr 11.4%

   \[\leadsto 4 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\pi, -0.25, \pi \cdot \left(u \cdot 0.5\right)\right)}\right)}^{3}} \]

9. Taylor expanded in u around 0 11.4%

   \[\leadsto 4 \cdot {\left(\sqrt[3]{\color{blue}{-0.25 \cdot \pi + 0.5 \cdot \left(u \cdot \pi\right)}}\right)}^{3} \]
10. Step-by-step derivation

    1. associate-*r* 11.4%

       \[\leadsto 4 \cdot {\left(\sqrt[3]{-0.25 \cdot \pi + \color{blue}{\left(0.5 \cdot u\right) \cdot \pi}}\right)}^{3} \]

    2. *-commutative 11.4%

       \[\leadsto 4 \cdot {\left(\sqrt[3]{-0.25 \cdot \pi + \color{blue}{\left(u \cdot 0.5\right)} \cdot \pi}\right)}^{3} \]

    3. distribute-rgt-in 11.4%

       \[\leadsto 4 \cdot {\left(\sqrt[3]{\color{blue}{\pi \cdot \left(-0.25 + u \cdot 0.5\right)}}\right)}^{3} \]

11. Simplified 11.4%

    \[\leadsto 4 \cdot {\left(\sqrt[3]{\color{blue}{\pi \cdot \left(-0.25 + u \cdot 0.5\right)}}\right)}^{3} \]

12. Final simplification 11.4%

    \[\leadsto 4 \cdot {\left(\sqrt[3]{\pi \cdot \left(-0.25 + u \cdot 0.5\right)}\right)}^{3} \]

Alternative 4: 11.6% accurate, 3.6× speedup

Math

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

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(Float32(u * 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 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around -inf 11.3%

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

   1. associate--r+ 11.3%

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

   2. cancel-sign-sub-inv 11.3%

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

   3. distribute-rgt-out-- 11.3%

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

   4. *-commutative 11.3%

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

   5. metadata-eval 11.3%

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

   6. metadata-eval 11.3%

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

   7. *-commutative 11.3%

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

6. Simplified 11.3%

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

7. Taylor expanded in u around 0 11.4%

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

8. Final simplification 11.4%

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

Alternative 5: 11.6% accurate, 6.9× 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%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in s around -inf 11.3%

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

   1. associate--r+ 11.3%

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

   2. cancel-sign-sub-inv 11.3%

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

   3. distribute-rgt-out-- 11.3%

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

   4. *-commutative 11.3%

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

   5. metadata-eval 11.3%

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

   6. metadata-eval 11.3%

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

   7. *-commutative 11.3%

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

6. Simplified 11.3%

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

7. Taylor expanded in s around 0 11.4%

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

   1. distribute-lft-in 11.4%

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

   2. *-commutative 11.4%

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

   3. associate-*r* 11.4%

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

   4. metadata-eval 11.4%

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

   5. *-commutative 11.4%

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

   6. associate-*r* 11.4%

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

   7. metadata-eval 11.4%

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

   8. associate-*r* 11.4%

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

   9. distribute-rgt-out 11.4%

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

   10. *-commutative 11.4%

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

9. Simplified 11.4%

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

10. Final simplification 11.4%

    \[\leadsto \pi \cdot \left(-1 + u \cdot 2\right) \]

Alternative 6: 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 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in u around 0 11.1%

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

   1. mul-1-neg 11.1%

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

6. Simplified 11.1%

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

7. Final simplification 11.1%

   \[\leadsto -\pi \]

Alternative 7: 4.6% accurate, 7.3× 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 98.8%

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

   1. distribute-lft-neg-out 98.8%

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

   2. distribute-rgt-neg-in 98.8%

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

   3. sub-neg 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in u around 0 11.1%

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

   1. expm1-log1p-u -0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(-\frac{\pi}{s}\right)\right)\right)} \]

   2. expm1-udef -0.0%

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

   3. add-sqr-sqrt -0.0%

      \[\leadsto e^{\mathsf{log1p}\left(s \cdot \color{blue}{\left(\sqrt{-\frac{\pi}{s}} \cdot \sqrt{-\frac{\pi}{s}}\right)}\right)} - 1 \]

   4. sqrt-unprod 3.3%

      \[\leadsto e^{\mathsf{log1p}\left(s \cdot \color{blue}{\sqrt{\left(-\frac{\pi}{s}\right) \cdot \left(-\frac{\pi}{s}\right)}}\right)} - 1 \]

   5. sqr-neg 3.3%

      \[\leadsto e^{\mathsf{log1p}\left(s \cdot \sqrt{\color{blue}{\frac{\pi}{s} \cdot \frac{\pi}{s}}}\right)} - 1 \]

   6. sqrt-unprod 4.5%

      \[\leadsto e^{\mathsf{log1p}\left(s \cdot \color{blue}{\left(\sqrt{\frac{\pi}{s}} \cdot \sqrt{\frac{\pi}{s}}\right)}\right)} - 1 \]

   7. add-sqr-sqrt 4.5%

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

6. Applied egg-rr 4.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(s \cdot \frac{\pi}{s}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 4.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \frac{\pi}{s}\right)\right)} \]

   2. expm1-log1p 4.5%

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

   3. associate-*r/ 4.6%

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

   4. associate-*l/ 4.6%

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

   5. *-inverses 4.6%

      \[\leadsto \color{blue}{1} \cdot \pi \]

   6. *-lft-identity 4.6%

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

8. Simplified 4.6%

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

9. Final simplification 4.6%

   \[\leadsto \pi \]

Reproduce

herbie shell --seed 2023255 
(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))))