Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 14.9s

Alternatives: 13

Speedup: 1.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 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, 1.3× 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(-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 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. Add Preprocessing

5. Final simplification 99.0%

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

Alternative 2: 25.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(0.25 + u \cdot 0.5\right)\\ s \cdot \left(\left(\log s - 0.25 \cdot \frac{s}{t\_0}\right) - \log \left(4 \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (* PI (+ 0.25 (* u 0.5)))))
   (* s (- (- (log s) (* 0.25 (/ s t_0))) (log (* 4.0 t_0))))))

C

float code(float u, float s) {
	float t_0 = ((float) M_PI) * (0.25f + (u * 0.5f));
	return s * ((logf(s) - (0.25f * (s / t_0))) - logf((4.0f * t_0)));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(pi) * Float32(Float32(0.25) + Float32(u * Float32(0.5))))
	return Float32(s * Float32(Float32(log(s) - Float32(Float32(0.25) * Float32(s / t_0))) - log(Float32(Float32(4.0) * t_0))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(pi) * (single(0.25) + (u * single(0.5)));
	tmp = s * ((log(s) - (single(0.25) * (s / t_0))) - log((single(4.0) * t_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. Add Preprocessing

5. Taylor expanded in s around inf 24.3%

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

   1. +-commutative 24.3%

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

   2. fma-define 24.3%

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

   3. associate--r+ 24.3%

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

   4. cancel-sign-sub-inv 24.3%

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

   5. distribute-rgt-out-- 24.3%

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

   6. *-commutative 24.3%

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

   7. metadata-eval 24.3%

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

   8. metadata-eval 24.3%

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

   9. *-commutative 24.3%

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

7. Simplified 24.3%

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

9. Applied egg-rr 24.6%

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

    1. fma-undefine 24.6%

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

    2. fma-undefine 24.6%

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

    3. distribute-lft-out 24.6%

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

    4. associate-+r+ 24.6%

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

    5. *-commutative 24.6%

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

    6. associate-*r* 24.6%

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

    7. *-commutative 24.6%

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

    8. distribute-lft-in 24.6%

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

    9. distribute-lft-out 24.6%

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

    10. metadata-eval 24.6%

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

    11. associate-*r* 24.6%

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

    12. *-commutative 24.6%

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

    13. metadata-eval 24.6%

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

    14. distribute-rgt-neg-in 24.6%

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

    15. associate-*r* 24.6%

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

    16. neg-sub0 24.6%

        \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\color{blue}{\left(0 - \pi \cdot \left(u \cdot 0.5\right)\right)} + \pi \cdot -0.25}{s}, 1\right)\right) \]

    17. associate--r- 24.6%

        \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\color{blue}{0 - \left(\pi \cdot \left(u \cdot 0.5\right) - \pi \cdot -0.25\right)}}{s}, 1\right)\right) \]

11. Simplified 24.6%

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

12. Taylor expanded in s around 0 24.8%

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

13. Final simplification 24.8%

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

Alternative 3: 25.2% accurate, 2.1× 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%

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

5. Taylor expanded in s around inf 24.3%

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

   1. +-commutative 24.3%

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

   2. fma-define 24.3%

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

   3. associate--r+ 24.3%

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

   4. cancel-sign-sub-inv 24.3%

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

   5. distribute-rgt-out-- 24.3%

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

   6. *-commutative 24.3%

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

   7. metadata-eval 24.3%

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

   8. metadata-eval 24.3%

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

   9. *-commutative 24.3%

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

7. Simplified 24.3%

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

8. Taylor expanded in u around 0 24.6%

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

9. Taylor expanded in s around 0 24.8%

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

    1. +-commutative 24.8%

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

    2. mul-1-neg 24.8%

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

    3. unsub-neg 24.8%

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

11. Simplified 24.8%

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

12. Final simplification 24.8%

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

Alternative 4: 25.2% accurate, 2.1× 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%

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

5. Taylor expanded in s around inf 24.3%

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

   1. +-commutative 24.3%

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

   2. fma-define 24.3%

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

   3. associate--r+ 24.3%

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

   4. cancel-sign-sub-inv 24.3%

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

   5. distribute-rgt-out-- 24.3%

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

   6. *-commutative 24.3%

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

   7. metadata-eval 24.3%

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

   8. metadata-eval 24.3%

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

   9. *-commutative 24.3%

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

7. Simplified 24.3%

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

8. Taylor expanded in u around 0 24.6%

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

9. Taylor expanded in s around 0 24.7%

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

    1. mul-1-neg 24.7%

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

    2. *-commutative 24.7%

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

    3. distribute-rgt-neg-in 24.7%

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

    4. mul-1-neg 24.7%

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

    5. unsub-neg 24.7%

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

11. Simplified 24.7%

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

12. Final simplification 24.7%

    \[\leadsto s \cdot \left(\log s - \log \pi\right) \]
13. Add Preprocessing

Alternative 5: 25.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(\pi \cdot \frac{-2}{1 - \frac{\pi}{s}}\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (- (* u (* PI (/ -2.0 (- 1.0 (/ PI s))))) (* s (log1p (/ PI s)))))

C

float code(float u, float s) {
	return (u * (((float) M_PI) * (-2.0f / (1.0f - (((float) M_PI) / s))))) - (s * log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(u * Float32(Float32(pi) * Float32(Float32(-2.0) / Float32(Float32(1.0) - Float32(Float32(pi) / s))))) - 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. Add Preprocessing

5. Taylor expanded in s around inf 24.3%

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

   1. +-commutative 24.3%

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

   2. fma-define 24.3%

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

   3. associate--r+ 24.3%

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

   4. cancel-sign-sub-inv 24.3%

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

   5. distribute-rgt-out-- 24.3%

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

   6. *-commutative 24.3%

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

   7. metadata-eval 24.3%

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

   8. metadata-eval 24.3%

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

   9. *-commutative 24.3%

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

7. Simplified 24.3%

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

9. Applied egg-rr 24.6%

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

    1. fma-undefine 24.6%

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

    2. fma-undefine 24.6%

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

    3. distribute-lft-out 24.6%

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

    4. associate-+r+ 24.6%

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

    5. *-commutative 24.6%

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

    6. associate-*r* 24.6%

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

    7. *-commutative 24.6%

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

    8. distribute-lft-in 24.6%

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

    9. distribute-lft-out 24.6%

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

    10. metadata-eval 24.6%

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

    11. associate-*r* 24.6%

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

    12. *-commutative 24.6%

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

    13. metadata-eval 24.6%

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

    14. distribute-rgt-neg-in 24.6%

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

    15. associate-*r* 24.6%

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

    16. neg-sub0 24.6%

        \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\color{blue}{\left(0 - \pi \cdot \left(u \cdot 0.5\right)\right)} + \pi \cdot -0.25}{s}, 1\right)\right) \]

    17. associate--r- 24.6%

        \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\color{blue}{0 - \left(\pi \cdot \left(u \cdot 0.5\right) - \pi \cdot -0.25\right)}}{s}, 1\right)\right) \]

11. Simplified 24.6%

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

12. Taylor expanded in u around 0 24.6%

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

    1. neg-mul-1 24.6%

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

    2. unsub-neg 24.6%

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

    3. *-commutative 24.6%

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

    4. associate-/l* 24.6%

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

    5. associate-*r* 24.6%

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

    6. *-commutative 24.6%

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

    7. associate-*r/ 24.6%

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

    8. *-rgt-identity 24.6%

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

    9. times-frac 24.6%

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

    10. +-commutative 24.6%

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

    11. /-rgt-identity 24.6%

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

    12. log1p-define 24.6%

        \[\leadsto u \cdot \left(\frac{-2}{\frac{\pi}{s} + 1} \cdot \pi\right) - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

14. Simplified 24.6%

    \[\leadsto \color{blue}{u \cdot \left(\frac{-2}{\frac{\pi}{s} + 1} \cdot \pi\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]
15. Step-by-step derivation

    1. add-sqr-sqrt 24.6%

       \[\leadsto u \cdot \left(\frac{-2}{\frac{\pi}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}} + 1} \cdot \pi\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

    2. sqrt-unprod 24.6%

       \[\leadsto u \cdot \left(\frac{-2}{\frac{\pi}{\color{blue}{\sqrt{s \cdot s}}} + 1} \cdot \pi\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

    3. sqr-neg 24.6%

       \[\leadsto u \cdot \left(\frac{-2}{\frac{\pi}{\sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}} + 1} \cdot \pi\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

    4. sqrt-unprod -0.0%

       \[\leadsto u \cdot \left(\frac{-2}{\frac{\pi}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} + 1} \cdot \pi\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

    5. add-sqr-sqrt 24.6%

       \[\leadsto u \cdot \left(\frac{-2}{\frac{\pi}{\color{blue}{-s}} + 1} \cdot \pi\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

    6. distribute-frac-neg2 24.6%

       \[\leadsto u \cdot \left(\frac{-2}{\color{blue}{\left(-\frac{\pi}{s}\right)} + 1} \cdot \pi\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

16. Applied egg-rr 24.6%

    \[\leadsto u \cdot \left(\frac{-2}{\color{blue}{\left(-\frac{\pi}{s}\right)} + 1} \cdot \pi\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

17. Final simplification 24.6%

    \[\leadsto u \cdot \left(\pi \cdot \frac{-2}{1 - \frac{\pi}{s}}\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
18. Add Preprocessing

Alternative 6: 25.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot -2\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \end{array} \]

FPCore

(FPCore (u s) :precision binary32 (- (* u (* s -2.0)) (* s (log1p (/ PI s)))))

C

float code(float u, float s) {
	return (u * (s * -2.0f)) - (s * log1pf((((float) M_PI) / s)));
}

Julia

function code(u, s)
	return Float32(Float32(u * Float32(s * Float32(-2.0))) - 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. Add Preprocessing

5. Taylor expanded in s around inf 24.3%

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

   1. +-commutative 24.3%

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

   2. fma-define 24.3%

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

   3. associate--r+ 24.3%

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

   4. cancel-sign-sub-inv 24.3%

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

   5. distribute-rgt-out-- 24.3%

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

   6. *-commutative 24.3%

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

   7. metadata-eval 24.3%

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

   8. metadata-eval 24.3%

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

   9. *-commutative 24.3%

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

7. Simplified 24.3%

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

9. Applied egg-rr 24.6%

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

    1. fma-undefine 24.6%

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

    2. fma-undefine 24.6%

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

    3. distribute-lft-out 24.6%

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

    4. associate-+r+ 24.6%

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

    5. *-commutative 24.6%

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

    6. associate-*r* 24.6%

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

    7. *-commutative 24.6%

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

    8. distribute-lft-in 24.6%

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

    9. distribute-lft-out 24.6%

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

    10. metadata-eval 24.6%

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

    11. associate-*r* 24.6%

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

    12. *-commutative 24.6%

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

    13. metadata-eval 24.6%

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

    14. distribute-rgt-neg-in 24.6%

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

    15. associate-*r* 24.6%

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

    16. neg-sub0 24.6%

        \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\color{blue}{\left(0 - \pi \cdot \left(u \cdot 0.5\right)\right)} + \pi \cdot -0.25}{s}, 1\right)\right) \]

    17. associate--r- 24.6%

        \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\color{blue}{0 - \left(\pi \cdot \left(u \cdot 0.5\right) - \pi \cdot -0.25\right)}}{s}, 1\right)\right) \]

11. Simplified 24.6%

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

12. Taylor expanded in u around 0 24.6%

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

    1. neg-mul-1 24.6%

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

    2. unsub-neg 24.6%

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

    3. *-commutative 24.6%

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

    4. associate-/l* 24.6%

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

    5. associate-*r* 24.6%

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

    6. *-commutative 24.6%

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

    7. associate-*r/ 24.6%

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

    8. *-rgt-identity 24.6%

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

    9. times-frac 24.6%

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

    10. +-commutative 24.6%

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

    11. /-rgt-identity 24.6%

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

    12. log1p-define 24.6%

        \[\leadsto u \cdot \left(\frac{-2}{\frac{\pi}{s} + 1} \cdot \pi\right) - s \cdot \color{blue}{\mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

14. Simplified 24.6%

    \[\leadsto \color{blue}{u \cdot \left(\frac{-2}{\frac{\pi}{s} + 1} \cdot \pi\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right)} \]

15. Taylor expanded in s around 0 24.6%

    \[\leadsto u \cdot \color{blue}{\left(-2 \cdot s\right)} - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]

16. Final simplification 24.6%

    \[\leadsto u \cdot \left(s \cdot -2\right) - s \cdot \mathsf{log1p}\left(\frac{\pi}{s}\right) \]
17. Add Preprocessing

Alternative 7: 25.1% accurate, 4.1× 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%

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

5. Taylor expanded in s around inf 24.3%

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

   1. +-commutative 24.3%

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

   2. fma-define 24.3%

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

   3. associate--r+ 24.3%

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

   4. cancel-sign-sub-inv 24.3%

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

   5. distribute-rgt-out-- 24.3%

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

   6. *-commutative 24.3%

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

   7. metadata-eval 24.3%

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

   8. metadata-eval 24.3%

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

   9. *-commutative 24.3%

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

7. Simplified 24.3%

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

9. Applied egg-rr 24.6%

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

    1. fma-undefine 24.6%

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

    2. fma-undefine 24.6%

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

    3. distribute-lft-out 24.6%

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

    4. associate-+r+ 24.6%

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

    5. *-commutative 24.6%

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

    6. associate-*r* 24.6%

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

    7. *-commutative 24.6%

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

    8. distribute-lft-in 24.6%

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

    9. distribute-lft-out 24.6%

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

    10. metadata-eval 24.6%

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

    11. associate-*r* 24.6%

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

    12. *-commutative 24.6%

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

    13. metadata-eval 24.6%

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

    14. distribute-rgt-neg-in 24.6%

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

    15. associate-*r* 24.6%

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

    16. neg-sub0 24.6%

        \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\color{blue}{\left(0 - \pi \cdot \left(u \cdot 0.5\right)\right)} + \pi \cdot -0.25}{s}, 1\right)\right) \]

    17. associate--r- 24.6%

        \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(-4, \frac{\color{blue}{0 - \left(\pi \cdot \left(u \cdot 0.5\right) - \pi \cdot -0.25\right)}}{s}, 1\right)\right) \]

11. Simplified 24.6%

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

12. Taylor expanded in u around 0 24.6%

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

    1. log1p-define 24.6%

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

14. Simplified 24.6%

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

15. Final simplification 24.6%

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

Alternative 8: 11.6% accurate, 25.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * (u * (((single(-0.25) * (single(pi) / u)) + (single(pi) * single(0.25))) - (single(pi) * 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. Add Preprocessing

5. Taylor expanded in s around inf 11.2%

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

6. Taylor expanded in u around inf 11.2%

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

7. Final simplification 11.2%

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

Alternative 9: 11.6% accurate, 33.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

5. Taylor expanded in s around inf 11.2%

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

   1. associate--r+ 11.2%

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

   2. cancel-sign-sub-inv 11.2%

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

   3. distribute-rgt-out-- 11.2%

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

   4. *-commutative 11.2%

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

   5. metadata-eval 11.2%

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

   6. metadata-eval 11.2%

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

   7. *-commutative 11.2%

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

7. Simplified 11.2%

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

8. Taylor expanded in u around inf 11.2%

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

9. Final simplification 11.2%

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

Alternative 10: 11.6% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(pi) * single(-0.25)) + (single(0.5) * (u * 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. Add Preprocessing

5. Taylor expanded in s around inf 11.2%

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

   1. associate--r+ 11.2%

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

   2. cancel-sign-sub-inv 11.2%

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

   3. distribute-rgt-out-- 11.2%

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

   4. *-commutative 11.2%

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

   5. metadata-eval 11.2%

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

   6. metadata-eval 11.2%

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

   7. *-commutative 11.2%

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

7. Simplified 11.2%

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

8. Final simplification 11.2%

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

Alternative 11: 11.6% accurate, 48.1× 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. Add Preprocessing

5. Taylor expanded in s around inf 11.2%

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

   1. associate--r+ 11.2%

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

   2. cancel-sign-sub-inv 11.2%

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

   3. distribute-rgt-out-- 11.2%

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

   4. *-commutative 11.2%

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

   5. metadata-eval 11.2%

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

   6. metadata-eval 11.2%

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

   7. *-commutative 11.2%

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

7. Simplified 11.2%

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

   1. associate-*l* 11.2%

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

   2. distribute-lft-out 11.2%

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

9. Applied egg-rr 11.2%

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

10. Final simplification 11.2%

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

Alternative 12: 11.4% accurate, 216.5× speedup

Math

\[\begin{array}{l} \\ -\pi \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

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. Add Preprocessing

5. Taylor expanded in u around 0 10.9%

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

   1. neg-mul-1 10.9%

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

7. Simplified 10.9%

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

8. Final simplification 10.9%

   \[\leadsto -\pi \]
9. Add Preprocessing

Alternative 13: 4.6% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ \pi \end{array} \]

FPCore

(FPCore (u s) :precision binary32 PI)

C

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

Julia

function code(u, s)
	return Float32(pi)
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

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. Add Preprocessing

5. Taylor expanded in s around inf 24.3%

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

   1. +-commutative 24.3%

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

   2. fma-define 24.3%

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

   3. associate--r+ 24.3%

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

   4. cancel-sign-sub-inv 24.3%

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

   5. distribute-rgt-out-- 24.3%

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

   6. *-commutative 24.3%

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

   7. metadata-eval 24.3%

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

   8. metadata-eval 24.3%

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

   9. *-commutative 24.3%

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

7. Simplified 24.3%

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

8. Taylor expanded in u around 0 24.6%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 8.9%

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

   3. sqr-neg 8.9%

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

   4. sqrt-unprod 7.6%

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

   5. add-sqr-sqrt 7.6%

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

   6. add-exp-log 7.6%

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

   7. fma-undefine 7.6%

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

   8. +-commutative 7.6%

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

   9. associate-/l* 7.6%

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

   10. associate-*r* 7.6%

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

   11. metadata-eval 7.6%

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

   12. *-un-lft-identity 7.6%

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

10. Applied egg-rr 7.6%

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

11. Taylor expanded in s around inf 4.8%

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

12. Final simplification 4.8%

    \[\leadsto \pi \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(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))))