Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 11.4s

Alternatives: 8

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.4× 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 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 25.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-def 24.5%

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

6. Simplified 24.5%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

9. Simplified 24.8%

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

10. Taylor expanded in s around 0 25.1%

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

    1. div-inv 25.1%

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

    2. unpow2 25.1%

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

    3. associate-*l* 25.1%

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

    4. div-inv 25.1%

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

    5. add-exp-log 25.1%

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

    6. add-exp-log 25.1%

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

    7. div-exp 25.1%

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

    8. remove-double-neg 25.1%

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

    9. neg-sub0 25.1%

       \[\leadsto -1 \cdot \left(s \cdot \left(\log \pi + -1 \cdot \log s\right)\right) + -1 \cdot \left(s \cdot e^{\color{blue}{\left(0 - \left(-\log s\right)\right)} - \log \pi}\right) \]

    10. associate--r+ 25.1%

        \[\leadsto -1 \cdot \left(s \cdot \left(\log \pi + -1 \cdot \log s\right)\right) + -1 \cdot \left(s \cdot e^{\color{blue}{0 - \left(\left(-\log s\right) + \log \pi\right)}}\right) \]

    11. +-commutative 25.1%

        \[\leadsto -1 \cdot \left(s \cdot \left(\log \pi + -1 \cdot \log s\right)\right) + -1 \cdot \left(s \cdot e^{0 - \color{blue}{\left(\log \pi + \left(-\log s\right)\right)}}\right) \]

    12. neg-sub0 25.1%

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

    13. add-sqr-sqrt -0.0%

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

    14. sqrt-unprod 10.5%

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

    15. sqr-neg 10.5%

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

12. Applied egg-rr 25.1%

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

13. Final simplification 25.1%

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

Alternative 3: 25.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-def 24.5%

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

6. Simplified 24.5%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

9. Simplified 24.8%

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

10. Taylor expanded in s around 0 25.1%

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

11. Final simplification 25.1%

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

Alternative 4: 25.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-def 24.5%

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

6. Simplified 24.5%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

9. Simplified 24.8%

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

10. Taylor expanded in s around 0 25.0%

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

    1. mul-1-neg 25.0%

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

12. Simplified 25.0%

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

13. Taylor expanded in s around 0 25.0%

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

14. Final simplification 25.0%

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

Alternative 5: 25.2% accurate, 3.6× speedup

Math

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

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(s * Float32(-log1p(Float32(Float32(pi) / s))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-def 24.5%

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

6. Simplified 24.5%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

9. Simplified 24.8%

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

10. Final simplification 24.8%

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

Alternative 6: 11.7% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 11.0%

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

   1. associate--r+ 11.0%

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

   2. cancel-sign-sub-inv 11.0%

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

   3. distribute-rgt-out-- 11.0%

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

   4. *-commutative 11.0%

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

   5. metadata-eval 11.0%

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

   6. metadata-eval 11.0%

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

   7. *-commutative 11.0%

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

6. Simplified 11.0%

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

   1. associate-*l* 11.0%

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

   2. distribute-lft-out 11.0%

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

8. Applied egg-rr 11.0%

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

9. Final simplification 11.0%

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

Alternative 7: 11.5% 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 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in u around 0 10.8%

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

   1. neg-mul-1 10.8%

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

6. Simplified 10.8%

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

7. Final simplification 10.8%

   \[\leadsto -\pi \]

Alternative 8: 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 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around -inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-def 24.5%

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

6. Simplified 24.5%

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

7. Taylor expanded in u around 0 24.8%

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

   1. log1p-def 24.8%

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

9. Simplified 24.8%

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

    1. add-cube-cbrt 24.8%

       \[\leadsto \color{blue}{\left(\sqrt[3]{s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)} \cdot \sqrt[3]{s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}\right) \cdot \sqrt[3]{s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}} \]

    2. pow3 24.8%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}\right)}^{3}} \]

    3. add-sqr-sqrt -0.0%

       \[\leadsto {\left(\sqrt[3]{s \cdot \color{blue}{\left(\sqrt{-\mathsf{log1p}\left(\frac{\pi}{s}\right)} \cdot \sqrt{-\mathsf{log1p}\left(\frac{\pi}{s}\right)}\right)}}\right)}^{3} \]

    4. sqrt-unprod 7.5%

       \[\leadsto {\left(\sqrt[3]{s \cdot \color{blue}{\sqrt{\left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right)}}}\right)}^{3} \]

    5. sqr-neg 7.5%

       \[\leadsto {\left(\sqrt[3]{s \cdot \sqrt{\color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right) \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)}}}\right)}^{3} \]

    6. sqrt-unprod 7.5%

       \[\leadsto {\left(\sqrt[3]{s \cdot \color{blue}{\left(\sqrt{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \cdot \sqrt{\mathsf{log1p}\left(\frac{\pi}{s}\right)}\right)}}\right)}^{3} \]

    7. add-sqr-sqrt 7.5%

       \[\leadsto {\left(\sqrt[3]{s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)}}\right)}^{3} \]

11. Applied egg-rr 7.5%

    \[\leadsto \color{blue}{{\left(\sqrt[3]{s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)}\right)}^{3}} \]

12. Taylor expanded in s around inf 4.7%

    \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \pi} \]
13. Step-by-step derivation

    1. pow-base-1 4.7%

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

    2. *-lft-identity 4.7%

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

14. Simplified 4.7%

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

15. Final simplification 4.7%

    \[\leadsto \pi \]

Reproduce

herbie shell --seed 2023334 
(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))))