Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 98.9%

Time: 14.9s

Alternatives: 14

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 25.3% 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-define 24.5%

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

   3. associate--r+ 24.5%

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

   4. cancel-sign-sub-inv 24.5%

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

   5. distribute-rgt-out-- 24.5%

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

   6. *-commutative 24.5%

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

   7. metadata-eval 24.5%

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

   8. metadata-eval 24.5%

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

   9. *-commutative 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in u around 0 24.7%

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

   1. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around 0 24.8%

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

    1. +-commutative 24.8%

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

    2. mul-1-neg 24.8%

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

    3. unsub-neg 24.8%

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

    4. mul-1-neg 24.8%

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

    5. distribute-neg-frac2 24.8%

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

    6. mul-1-neg 24.8%

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

    7. unsub-neg 24.8%

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

13. Simplified 24.8%

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

14. Taylor expanded in s around 0 24.8%

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

    1. neg-mul-1 24.8%

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

    2. unsub-neg 24.8%

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

16. Simplified 24.8%

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

17. Final simplification 24.8%

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

Alternative 3: 25.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-define 24.5%

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

   3. associate--r+ 24.5%

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

   4. cancel-sign-sub-inv 24.5%

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

   5. distribute-rgt-out-- 24.5%

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

   6. *-commutative 24.5%

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

   7. metadata-eval 24.5%

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

   8. metadata-eval 24.5%

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

   9. *-commutative 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in u around 0 24.7%

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

   1. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around 0 24.8%

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

    1. +-commutative 24.8%

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

    2. mul-1-neg 24.8%

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

    3. unsub-neg 24.8%

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

    4. mul-1-neg 24.8%

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

    5. distribute-neg-frac2 24.8%

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

    6. mul-1-neg 24.8%

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

    7. unsub-neg 24.8%

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

13. Simplified 24.8%

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

14. Final simplification 24.8%

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

Alternative 4: 25.3% 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-define 24.5%

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

   3. associate--r+ 24.5%

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

   4. cancel-sign-sub-inv 24.5%

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

   5. distribute-rgt-out-- 24.5%

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

   6. *-commutative 24.5%

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

   7. metadata-eval 24.5%

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

   8. metadata-eval 24.5%

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

   9. *-commutative 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in u around 0 24.7%

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

   1. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around 0 24.8%

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

    1. +-commutative 24.8%

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

    2. mul-1-neg 24.8%

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

    3. unsub-neg 24.8%

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

    4. mul-1-neg 24.8%

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

    5. distribute-neg-frac2 24.8%

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

    6. mul-1-neg 24.8%

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

    7. unsub-neg 24.8%

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

13. Simplified 24.8%

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

14. Taylor expanded in s around 0 24.8%

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

15. Final simplification 24.8%

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

Alternative 5: 25.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-define 24.5%

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

   3. associate--r+ 24.5%

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

   4. cancel-sign-sub-inv 24.5%

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

   5. distribute-rgt-out-- 24.5%

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

   6. *-commutative 24.5%

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

   7. metadata-eval 24.5%

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

   8. metadata-eval 24.5%

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

   9. *-commutative 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in u around 0 24.7%

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

   1. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around 0 24.8%

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

    1. +-commutative 24.8%

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

    2. mul-1-neg 24.8%

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

    3. unsub-neg 24.8%

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

    4. mul-1-neg 24.8%

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

    5. distribute-neg-frac2 24.8%

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

    6. mul-1-neg 24.8%

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

    7. unsub-neg 24.8%

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

13. Simplified 24.8%

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

    1. pow1 24.8%

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

15. Applied egg-rr 24.8%

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

    1. unpow1 24.8%

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

17. Simplified 24.8%

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

18. Final simplification 24.8%

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

Alternative 6: 25.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-define 24.5%

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

   3. associate--r+ 24.5%

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

   4. cancel-sign-sub-inv 24.5%

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

   5. distribute-rgt-out-- 24.5%

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

   6. *-commutative 24.5%

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

   7. metadata-eval 24.5%

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

   8. metadata-eval 24.5%

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

   9. *-commutative 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in u around 0 24.7%

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

   1. *-commutative 24.7%

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

10. Simplified 24.7%

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

    1. *-un-lft-identity 24.7%

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

    2. log-prod 24.7%

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

    3. metadata-eval 24.7%

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

    4. associate-/l* 24.7%

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

12. Applied egg-rr 24.7%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\left(0 + \log \left(\mathsf{fma}\left(-4, \pi \cdot \frac{-0.25}{s}, 1\right)\right)\right)} \]
13. Step-by-step derivation

    1. +-lft-identity 24.7%

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

    2. fma-undefine 24.7%

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

    3. *-commutative 24.7%

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

    4. associate-*l/ 24.7%

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

    5. associate-*r/ 24.7%

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

    6. *-rgt-identity 24.7%

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

    7. associate-*r/ 24.7%

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

    8. associate-*r* 24.7%

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

    9. metadata-eval 24.7%

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

    10. associate-*l* 24.7%

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

    11. *-lft-identity 24.7%

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

    12. +-commutative 24.7%

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

    13. associate-*r/ 24.7%

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

    14. *-rgt-identity 24.7%

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

14. Simplified 24.7%

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

15. Final simplification 24.7%

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

Alternative 7: 25.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-define 24.5%

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

   3. associate--r+ 24.5%

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

   4. cancel-sign-sub-inv 24.5%

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

   5. distribute-rgt-out-- 24.5%

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

   6. *-commutative 24.5%

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

   7. metadata-eval 24.5%

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

   8. metadata-eval 24.5%

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

   9. *-commutative 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in u around 0 24.7%

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

   1. *-commutative 24.7%

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

10. Simplified 24.7%

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

    1. *-un-lft-identity 24.7%

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

    2. log-prod 24.7%

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

    3. metadata-eval 24.7%

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

    4. associate-/l* 24.7%

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

12. Applied egg-rr 24.7%

    \[\leadsto \left(-s\right) \cdot \color{blue}{\left(0 + \log \left(\mathsf{fma}\left(-4, \pi \cdot \frac{-0.25}{s}, 1\right)\right)\right)} \]
13. Step-by-step derivation

    1. +-lft-identity 24.7%

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

    2. fma-undefine 24.7%

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

    3. *-commutative 24.7%

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

    4. associate-*l/ 24.7%

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

    5. associate-*r/ 24.7%

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

    6. *-rgt-identity 24.7%

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

    7. associate-*r/ 24.7%

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

    8. associate-*r* 24.7%

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

    9. metadata-eval 24.7%

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

    10. associate-*l* 24.7%

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

    11. *-lft-identity 24.7%

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

    12. +-commutative 24.7%

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

    13. log1p-define 24.7%

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

    14. associate-*r/ 24.7%

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

    15. *-rgt-identity 24.7%

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

14. Simplified 24.7%

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

15. Final simplification 24.7%

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

Alternative 8: 12.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{{s}^{2}}{-\pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-define 24.5%

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

   3. associate--r+ 24.5%

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

   4. cancel-sign-sub-inv 24.5%

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

   5. distribute-rgt-out-- 24.5%

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

   6. *-commutative 24.5%

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

   7. metadata-eval 24.5%

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

   8. metadata-eval 24.5%

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

   9. *-commutative 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in u around 0 24.7%

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

   1. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around 0 24.8%

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

    1. +-commutative 24.8%

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

    2. mul-1-neg 24.8%

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

    3. unsub-neg 24.8%

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

    4. mul-1-neg 24.8%

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

    5. distribute-neg-frac2 24.8%

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

    6. mul-1-neg 24.8%

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

    7. unsub-neg 24.8%

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

13. Simplified 24.8%

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

14. Taylor expanded in s around inf 12.9%

    \[\leadsto \color{blue}{-1 \cdot \frac{{s}^{2}}{\pi}} \]
15. Step-by-step derivation

    1. mul-1-neg 12.9%

       \[\leadsto \color{blue}{-\frac{{s}^{2}}{\pi}} \]

    2. distribute-neg-frac2 12.9%

       \[\leadsto \color{blue}{\frac{{s}^{2}}{-\pi}} \]

16. Simplified 12.9%

    \[\leadsto \color{blue}{\frac{{s}^{2}}{-\pi}} \]

17. Final simplification 12.9%

    \[\leadsto \frac{{s}^{2}}{-\pi} \]
18. Add Preprocessing

Alternative 9: 11.6% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) + u \cdot \left(0.25 \cdot \left(\pi + \pi \cdot \frac{-1}{u}\right)\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* 4.0 (+ (* 0.25 (* u PI)) (* u (* 0.25 (+ PI (* PI (/ -1.0 u))))))))

C

float code(float u, float s) {
	return 4.0f * ((0.25f * (u * ((float) M_PI))) + (u * (0.25f * (((float) M_PI) + (((float) M_PI) * (-1.0f / u))))));
}

Julia

function code(u, s)
	return Float32(Float32(4.0) * Float32(Float32(Float32(0.25) * Float32(u * Float32(pi))) + Float32(u * Float32(Float32(0.25) * Float32(Float32(pi) + Float32(Float32(pi) * Float32(Float32(-1.0) / u)))))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(4.0) * ((single(0.25) * (u * single(pi))) + (u * (single(0.25) * (single(pi) + (single(pi) * (single(-1.0) / u))))));
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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. Taylor expanded in u around inf 11.4%

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

   1. +-commutative 11.4%

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

   2. metadata-eval 11.4%

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

   3. cancel-sign-sub-inv 11.4%

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

   4. distribute-lft-out-- 11.4%

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

8. Simplified 11.4%

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

   1. div-inv 11.5%

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

10. Applied egg-rr 11.5%

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

11. Final simplification 11.5%

    \[\leadsto 4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) + u \cdot \left(0.25 \cdot \left(\pi + \pi \cdot \frac{-1}{u}\right)\right)\right) \]
12. Add Preprocessing

Alternative 10: 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 99.1%

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

3. Simplified 99.1%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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. Taylor expanded in u around inf 11.4%

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

7. Final simplification 11.4%

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

Alternative 11: 11.6% accurate, 33.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
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 inf 11.4%

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

9. Final simplification 11.4%

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

Alternative 12: 11.6% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

   1. clear-num 99.1%

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

   2. associate-/r/ 99.1%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in s around inf 11.4%

   \[\leadsto \color{blue}{4 \cdot \left(0.25 \cdot \left(u \cdot \pi\right) - \left(-0.25 \cdot \left(u \cdot \pi\right) + 0.25 \cdot \pi\right)\right)} \]
8. 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. *-commutative 11.4%

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

   4. *-commutative 11.4%

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

   5. distribute-rgt-out-- 11.4%

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

   6. metadata-eval 11.4%

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

   7. *-commutative 11.4%

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

   8. *-commutative 11.4%

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

   9. metadata-eval 11.4%

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

   10. +-commutative 11.4%

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

   11. associate-*r* 11.4%

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

   12. *-commutative 11.4%

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

   13. distribute-rgt-out 11.4%

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

9. Simplified 11.4%

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

10. Final simplification 11.4%

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

Alternative 13: 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 99.1%

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

3. Simplified 99.1%

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

Alternative 14: 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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 24.5%

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

   1. +-commutative 24.5%

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

   2. fma-define 24.5%

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

   3. associate--r+ 24.5%

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

   4. cancel-sign-sub-inv 24.5%

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

   5. distribute-rgt-out-- 24.5%

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

   6. *-commutative 24.5%

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

   7. metadata-eval 24.5%

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

   8. metadata-eval 24.5%

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

   9. *-commutative 24.5%

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

7. Simplified 24.5%

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

8. Taylor expanded in u around 0 24.7%

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

   1. *-commutative 24.7%

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

10. Simplified 24.7%

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

11. Taylor expanded in s around inf 11.1%

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

    1. associate-*r/ 11.1%

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

    2. add-sqr-sqrt -0.0%

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

    3. sqrt-unprod 8.6%

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

    4. sqr-neg 8.6%

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

    5. sqrt-unprod 4.8%

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

    6. add-sqr-sqrt 4.8%

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

13. Applied egg-rr 4.8%

    \[\leadsto \color{blue}{\frac{s \cdot \pi}{s}} \]
14. Step-by-step derivation

    1. *-commutative 4.8%

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

    2. associate-/l* 4.8%

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

    3. *-inverses 4.8%

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

    4. *-rgt-identity 4.8%

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

15. Simplified 4.8%

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

16. Final simplification 4.8%

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

Reproduce

herbie shell --seed 2024075 
(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))))