Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 34.1s

Alternatives: 9

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.4× speedup

Math

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

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(Float32(-s) * 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%

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

4. Final simplification 99.0%

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

Alternative 2: 25.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, s \cdot u, \mathsf{fma}\left(2, s \cdot \left(u \cdot u\right), 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)\right) - s \cdot \log \left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (-
  (fma
   2.0
   (* s u)
   (fma 2.0 (* s (* u u)) (* 2.6666666666666665 (* s (pow u 3.0)))))
  (* s (log (/ PI s)))))

C

float code(float u, float s) {
	return fmaf(2.0f, (s * u), fmaf(2.0f, (s * (u * u)), (2.6666666666666665f * (s * powf(u, 3.0f))))) - (s * logf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(fma(Float32(2.0), Float32(s * u), fma(Float32(2.0), Float32(s * Float32(u * u)), Float32(Float32(2.6666666666666665) * Float32(s * (u ^ Float32(3.0)))))) - Float32(s * log(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%

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

4. Taylor expanded in s around -inf 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

6. Simplified 24.7%

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

   1. expm1-log1p-u 24.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}, 1\right)\right)\right)\right)} \]

   2. expm1-udef 14.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1} \]

   3. distribute-lft-out 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\color{blue}{\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) + u \cdot -0.25\right)}}{s}, 1\right)\right)\right)} - 1 \]

   4. fma-def 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\color{blue}{\mathsf{fma}\left(u, -0.25, 0.25\right)} + u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1 \]

8. Applied egg-rr 14.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\mathsf{fma}\left(u, -0.25, 0.25\right) + u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 24.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\mathsf{fma}\left(u, -0.25, 0.25\right) + u \cdot -0.25\right)}{s}, 1\right)\right)\right)\right)} \]

   2. expm1-log1p 24.7%

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

   3. distribute-lft-neg-in 24.7%

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

   4. distribute-rgt-neg-in 24.7%

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

   5. associate-/l* 24.7%

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

   6. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around 0 24.7%

    \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\log \left(4 \cdot \left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)\right) + -1 \cdot \log s\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* 24.7%

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

    2. neg-mul-1 24.7%

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

    3. *-commutative 24.7%

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

    4. mul-1-neg 24.7%

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

13. Simplified 24.7%

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

14. Taylor expanded in u around 0 25.0%

    \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\log \pi - \log s\right)\right) + \left(2 \cdot \left(s \cdot u\right) + \left(2 \cdot \left(s \cdot {u}^{2}\right) + 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)\right)} \]
15. Step-by-step derivation

    1. +-commutative 25.0%

       \[\leadsto \color{blue}{\left(2 \cdot \left(s \cdot u\right) + \left(2 \cdot \left(s \cdot {u}^{2}\right) + 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)\right) + -1 \cdot \left(s \cdot \left(\log \pi - \log s\right)\right)} \]

    2. mul-1-neg 25.0%

       \[\leadsto \left(2 \cdot \left(s \cdot u\right) + \left(2 \cdot \left(s \cdot {u}^{2}\right) + 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)\right) + \color{blue}{\left(-s \cdot \left(\log \pi - \log s\right)\right)} \]

    3. unsub-neg 25.0%

       \[\leadsto \color{blue}{\left(2 \cdot \left(s \cdot u\right) + \left(2 \cdot \left(s \cdot {u}^{2}\right) + 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)\right) - s \cdot \left(\log \pi - \log s\right)} \]

    4. fma-def 25.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, s \cdot u, 2 \cdot \left(s \cdot {u}^{2}\right) + 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)} - s \cdot \left(\log \pi - \log s\right) \]

    5. fma-def 25.0%

       \[\leadsto \mathsf{fma}\left(2, s \cdot u, \color{blue}{\mathsf{fma}\left(2, s \cdot {u}^{2}, 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)}\right) - s \cdot \left(\log \pi - \log s\right) \]

    6. unpow2 25.0%

       \[\leadsto \mathsf{fma}\left(2, s \cdot u, \mathsf{fma}\left(2, s \cdot \color{blue}{\left(u \cdot u\right)}, 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)\right) - s \cdot \left(\log \pi - \log s\right) \]

    7. log-div 25.0%

       \[\leadsto \mathsf{fma}\left(2, s \cdot u, \mathsf{fma}\left(2, s \cdot \left(u \cdot u\right), 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)\right) - s \cdot \color{blue}{\log \left(\frac{\pi}{s}\right)} \]

16. Simplified 25.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(2, s \cdot u, \mathsf{fma}\left(2, s \cdot \left(u \cdot u\right), 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)\right) - s \cdot \log \left(\frac{\pi}{s}\right)} \]

17. Final simplification 25.0%

    \[\leadsto \mathsf{fma}\left(2, s \cdot u, \mathsf{fma}\left(2, s \cdot \left(u \cdot u\right), 2.6666666666666665 \cdot \left(s \cdot {u}^{3}\right)\right)\right) - s \cdot \log \left(\frac{\pi}{s}\right) \]

Alternative 3: 25.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Taylor expanded in s around -inf 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

6. Simplified 24.7%

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

   1. expm1-log1p-u 24.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}, 1\right)\right)\right)\right)} \]

   2. expm1-udef 14.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1} \]

   3. distribute-lft-out 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\color{blue}{\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) + u \cdot -0.25\right)}}{s}, 1\right)\right)\right)} - 1 \]

   4. fma-def 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\color{blue}{\mathsf{fma}\left(u, -0.25, 0.25\right)} + u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1 \]

8. Applied egg-rr 14.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\mathsf{fma}\left(u, -0.25, 0.25\right) + u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 24.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\mathsf{fma}\left(u, -0.25, 0.25\right) + u \cdot -0.25\right)}{s}, 1\right)\right)\right)\right)} \]

   2. expm1-log1p 24.7%

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

   3. distribute-lft-neg-in 24.7%

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

   4. distribute-rgt-neg-in 24.7%

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

   5. associate-/l* 24.7%

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

   6. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around 0 24.7%

    \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\log \left(4 \cdot \left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)\right) + -1 \cdot \log s\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* 24.7%

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

    2. neg-mul-1 24.7%

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

    3. *-commutative 24.7%

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

    4. mul-1-neg 24.7%

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

13. Simplified 24.7%

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

14. Taylor expanded in u around 0 25.0%

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

    1. fma-def 25.0%

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

    2. distribute-lft-out 25.0%

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

    3. unpow2 25.0%

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

16. Simplified 25.0%

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

17. Final simplification 25.0%

    \[\leadsto \mathsf{fma}\left(-1, s \cdot \left(\log \pi - \log s\right), 2 \cdot \left(s \cdot u + s \cdot \left(u \cdot u\right)\right)\right) \]

Alternative 4: 25.2% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * ((s * u) + (s * (u * u)))) - (s * log((single(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%

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

4. Taylor expanded in s around -inf 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

6. Simplified 24.7%

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

   1. expm1-log1p-u 24.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}, 1\right)\right)\right)\right)} \]

   2. expm1-udef 14.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1} \]

   3. distribute-lft-out 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\color{blue}{\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) + u \cdot -0.25\right)}}{s}, 1\right)\right)\right)} - 1 \]

   4. fma-def 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\color{blue}{\mathsf{fma}\left(u, -0.25, 0.25\right)} + u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1 \]

8. Applied egg-rr 14.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\mathsf{fma}\left(u, -0.25, 0.25\right) + u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 24.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\mathsf{fma}\left(u, -0.25, 0.25\right) + u \cdot -0.25\right)}{s}, 1\right)\right)\right)\right)} \]

   2. expm1-log1p 24.7%

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

   3. distribute-lft-neg-in 24.7%

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

   4. distribute-rgt-neg-in 24.7%

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

   5. associate-/l* 24.7%

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

   6. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around 0 24.7%

    \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\log \left(4 \cdot \left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)\right) + -1 \cdot \log s\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* 24.7%

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

    2. neg-mul-1 24.7%

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

    3. *-commutative 24.7%

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

    4. mul-1-neg 24.7%

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

13. Simplified 24.7%

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

14. Taylor expanded in u around 0 25.0%

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

    1. +-commutative 25.0%

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

    2. mul-1-neg 25.0%

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

    3. unsub-neg 25.0%

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

    4. distribute-lft-out 25.0%

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

    5. unpow2 25.0%

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

    6. log-div 25.0%

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

16. Simplified 25.0%

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

17. Final simplification 25.0%

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

Alternative 5: 25.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * (s * u)) - (s * log((single(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%

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

4. Taylor expanded in s around -inf 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

6. Simplified 24.7%

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

   1. expm1-log1p-u 24.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}, 1\right)\right)\right)\right)} \]

   2. expm1-udef 14.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1} \]

   3. distribute-lft-out 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\color{blue}{\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) + u \cdot -0.25\right)}}{s}, 1\right)\right)\right)} - 1 \]

   4. fma-def 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\color{blue}{\mathsf{fma}\left(u, -0.25, 0.25\right)} + u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1 \]

8. Applied egg-rr 14.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\mathsf{fma}\left(u, -0.25, 0.25\right) + u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 24.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\mathsf{fma}\left(u, -0.25, 0.25\right) + u \cdot -0.25\right)}{s}, 1\right)\right)\right)\right)} \]

   2. expm1-log1p 24.7%

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

   3. distribute-lft-neg-in 24.7%

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

   4. distribute-rgt-neg-in 24.7%

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

   5. associate-/l* 24.7%

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

   6. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around 0 24.7%

    \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\log \left(4 \cdot \left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)\right) + -1 \cdot \log s\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* 24.7%

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

    2. neg-mul-1 24.7%

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

    3. *-commutative 24.7%

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

    4. mul-1-neg 24.7%

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

13. Simplified 24.7%

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

14. Taylor expanded in u around 0 24.9%

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

    1. +-commutative 24.9%

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

    2. mul-1-neg 24.9%

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

    3. unsub-neg 24.9%

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

    4. log-div 24.9%

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

16. Simplified 24.9%

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

17. Final simplification 24.9%

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

Alternative 6: 25.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = -s * log((single(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%

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

4. Taylor expanded in s around -inf 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

6. Simplified 24.7%

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

   1. expm1-log1p-u 24.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}, 1\right)\right)\right)\right)} \]

   2. expm1-udef 14.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(u \cdot -0.25 + 0.25\right) + \pi \cdot \left(u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1} \]

   3. distribute-lft-out 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\color{blue}{\pi \cdot \left(\left(u \cdot -0.25 + 0.25\right) + u \cdot -0.25\right)}}{s}, 1\right)\right)\right)} - 1 \]

   4. fma-def 14.7%

      \[\leadsto e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\color{blue}{\mathsf{fma}\left(u, -0.25, 0.25\right)} + u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1 \]

8. Applied egg-rr 14.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\mathsf{fma}\left(u, -0.25, 0.25\right) + u \cdot -0.25\right)}{s}, 1\right)\right)\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 24.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-s\right) \cdot \log \left(\mathsf{fma}\left(4, \frac{\pi \cdot \left(\mathsf{fma}\left(u, -0.25, 0.25\right) + u \cdot -0.25\right)}{s}, 1\right)\right)\right)\right)} \]

   2. expm1-log1p 24.7%

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

   3. distribute-lft-neg-in 24.7%

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

   4. distribute-rgt-neg-in 24.7%

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

   5. associate-/l* 24.7%

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

   6. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around 0 24.7%

    \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\log \left(4 \cdot \left(\pi \cdot \left(0.25 + -0.5 \cdot u\right)\right)\right) + -1 \cdot \log s\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* 24.7%

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

    2. neg-mul-1 24.7%

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

    3. *-commutative 24.7%

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

    4. mul-1-neg 24.7%

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

13. Simplified 24.7%

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

14. Taylor expanded in u around 0 24.9%

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

    1. mul-1-neg 24.9%

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

    2. log-div 24.9%

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

    3. distribute-rgt-neg-in 24.9%

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

16. Simplified 24.9%

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

17. Final simplification 24.9%

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

Alternative 7: 25.2% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (u s) :precision binary32 (* (- s) (log1p (/ PI s))))

C

float code(float u, float s) {
	return -s * log1pf((((float) M_PI) / s));
}

Julia

function code(u, s)
	return Float32(Float32(-s) * log1p(Float32(Float32(pi) / s)))
end

Derivation

1. Initial program 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%

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

4. Taylor expanded in s around -inf 24.7%

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

   1. +-commutative 24.7%

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

   2. fma-def 24.7%

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

6. Simplified 24.7%

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

7. Taylor expanded in u around 0 24.9%

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

   1. log1p-def 24.9%

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

   2. associate-*r* 24.9%

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

   3. neg-mul-1 24.9%

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

9. Simplified 24.9%

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

10. Final simplification 24.9%

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

Alternative 8: 11.5% 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%

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

4. Taylor expanded in s around inf 11.1%

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

   1. associate--r+ 11.1%

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

   2. cancel-sign-sub-inv 11.1%

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

   3. distribute-rgt-out-- 11.1%

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

   4. *-commutative 11.1%

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

   5. metadata-eval 11.1%

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

   6. metadata-eval 11.1%

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

   7. *-commutative 11.1%

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

6. Simplified 11.1%

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

7. Taylor expanded in u around 0 11.1%

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

   1. +-commutative 11.1%

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

   2. associate-*r* 11.1%

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

   3. *-commutative 11.1%

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

   4. distribute-rgt-out 11.1%

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

   5. *-commutative 11.1%

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

9. Simplified 11.1%

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

10. Final simplification 11.1%

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

Alternative 9: 11.3% 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%

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

4. Taylor expanded in u around 0 10.9%

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

   1. neg-mul-1 10.9%

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

6. Simplified 10.9%

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

7. Final simplification 10.9%

   \[\leadsto -\pi \]

Reproduce

herbie shell --seed 2023275 
(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))))