Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 19.0s

Alternatives: 9

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

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

FPCore

(FPCore (u s)
 :precision binary32
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ PI (- s)))))
       (/ (- 1.0 u) (+ 1.0 (exp (/ 1.0 (/ s PI)))))))
     -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((1.0f / (s / ((float) M_PI)))))))) + -1.0f));
}

Julia

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

   1. clear-num 99.0%

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

   2. inv-pow 99.0%

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

6. Applied egg-rr 99.0%

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

   1. unpow-1 99.0%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(-1.0) + 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))))))))))
end

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 3: 25.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

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

   1. mul-1-neg 24.6%

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

   2. log1p-define 24.6%

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

   3. distribute-rgt-neg-in 24.6%

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

10. Simplified 24.6%

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

    1. add-cube-cbrt 24.6%

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

    2. pow3 24.6%

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

12. Applied egg-rr 24.6%

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

13. Taylor expanded in s around 0 24.7%

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

    1. mul-1-neg 24.7%

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

    2. unsub-neg 24.7%

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

    3. mul-1-neg 24.7%

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

    4. distribute-rgt-neg-in 24.7%

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

    5. mul-1-neg 24.7%

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

    6. sub-neg 24.7%

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

    7. log-div 24.6%

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

    8. log-rec 24.6%

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

    9. associate-/r/ 24.7%

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

    10. associate-*l/ 24.7%

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

    11. *-lft-identity 24.7%

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

15. Simplified 24.7%

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

16. Final simplification 24.7%

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

Alternative 4: 25.4% 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 98.9%

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

3. Simplified 98.9%

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

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

   1. mul-1-neg 24.6%

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

   2. log1p-define 24.6%

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

   3. distribute-rgt-neg-in 24.6%

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

10. Simplified 24.6%

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

11. Taylor expanded in s around 0 24.7%

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

12. Final simplification 24.7%

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

Alternative 5: 25.4% 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 98.9%

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

3. Simplified 98.9%

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

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

   1. mul-1-neg 24.6%

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

   2. log1p-define 24.6%

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

   3. distribute-rgt-neg-in 24.6%

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

10. Simplified 24.6%

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

11. Taylor expanded in s around 0 24.7%

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

    1. mul-1-neg 24.7%

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

13. Simplified 24.7%

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

14. Final simplification 24.7%

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

Alternative 6: 25.4% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

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

   1. mul-1-neg 24.6%

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

   2. log1p-define 24.6%

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

   3. distribute-rgt-neg-in 24.6%

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

10. Simplified 24.6%

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

    1. add-cube-cbrt 24.6%

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

    2. pow3 24.6%

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

12. Applied egg-rr 24.6%

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

13. Taylor expanded in s around 0 24.7%

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

    1. mul-1-neg 24.7%

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

    2. distribute-rgt-neg-in 24.7%

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

    3. mul-1-neg 24.7%

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

    4. sub-neg 24.7%

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

    5. log-div 24.6%

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

    6. log-rec 24.6%

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

    7. associate-/r/ 24.7%

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

    8. associate-*l/ 24.7%

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

    9. *-lft-identity 24.7%

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

15. Simplified 24.7%

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

16. Final simplification 24.7%

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

Alternative 7: 11.5% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around inf 11.4%

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

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. distribute-rgt-out-- 11.4%

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

   4. *-commutative 11.4%

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

   5. metadata-eval 11.4%

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

   6. metadata-eval 11.4%

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

   7. *-commutative 11.4%

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

7. Simplified 11.4%

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

8. Final simplification 11.4%

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

Alternative 8: 11.5% 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in s around -inf 11.4%

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

   1. associate--r+ 11.4%

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

   2. cancel-sign-sub-inv 11.4%

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

   3. metadata-eval 11.4%

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

   4. cancel-sign-sub-inv 11.4%

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

   5. associate-*r* 11.4%

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

   6. distribute-rgt-out 11.4%

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

   7. *-commutative 11.4%

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

   8. metadata-eval 11.4%

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

   9. *-commutative 11.4%

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

   10. associate-*l* 11.4%

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

7. Simplified 11.4%

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

8. Taylor expanded in u around 0 11.4%

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

   1. associate-*r* 11.4%

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

   2. distribute-rgt-out 11.4%

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

10. Simplified 11.4%

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

11. Final simplification 11.4%

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

Alternative 9: 11.3% 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 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.1%

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

   1. neg-mul-1 11.1%

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

7. Simplified 11.1%

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

8. Final simplification 11.1%

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

Reproduce

herbie shell --seed 2024052 
(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))))