Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 14.6s

Alternatives: 15

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. Add Preprocessing

Alternative 3: 25.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * (((log(s) - (u * single(-2.0))) - log(single(pi))) - (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 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. associate-*r/ 25.1%

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

   4. associate-*r/ 25.1%

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

   5. div-sub 25.1%

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

   6. distribute-rgt-out-- 25.1%

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

   7. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot \color{blue}{-0.5}}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot -0.5}{s}\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0 25.1%

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

10. Taylor expanded in s around 0 25.1%

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

11. Final simplification 25.1%

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

Alternative 4: 25.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * (((log(s) - (s / single(pi))) - log(single(pi))) - (u * single(-2.0)));
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. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. associate-*r/ 25.1%

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

   4. associate-*r/ 25.1%

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

   5. div-sub 25.1%

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

   6. distribute-rgt-out-- 25.1%

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

   7. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot \color{blue}{-0.5}}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot -0.5}{s}\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0 25.1%

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

10. Taylor expanded in s around 0 25.1%

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

    1. neg-mul-1 25.1%

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

    2. +-commutative 25.1%

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

    3. unsub-neg 25.1%

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

12. Simplified 25.1%

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

13. Final simplification 25.1%

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

Alternative 5: 25.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = (single(2.0) * (s * u)) + (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 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. associate-*r/ 25.1%

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

   4. associate-*r/ 25.1%

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

   5. div-sub 25.1%

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

   6. distribute-rgt-out-- 25.1%

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

   7. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot \color{blue}{-0.5}}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot -0.5}{s}\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0 25.1%

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

    1. mul-1-neg 25.1%

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

    2. distribute-rgt-neg-in 25.1%

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

    3. mul-1-neg 25.1%

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

    4. distribute-lft-in 25.1%

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

    5. fma-define 25.1%

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

    6. mul-1-neg 25.1%

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

    7. fmm-undef 25.1%

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

    8. mul-1-neg 25.1%

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

    9. mul-1-neg 25.1%

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

    10. unsub-neg 25.1%

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

11. Simplified 25.1%

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

12. Taylor expanded in u around 0 25.1%

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

Alternative 6: 25.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u, s)
	tmp = s * (log(s) - (log(single(pi)) + (u * single(-2.0))));
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. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. associate-*r/ 25.1%

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

   4. associate-*r/ 25.1%

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

   5. div-sub 25.1%

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

   6. distribute-rgt-out-- 25.1%

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

   7. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot \color{blue}{-0.5}}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot -0.5}{s}\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0 25.1%

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

    1. mul-1-neg 25.1%

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

    2. distribute-rgt-neg-in 25.1%

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

    3. mul-1-neg 25.1%

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

    4. distribute-lft-in 25.1%

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

    5. fma-define 25.1%

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

    6. mul-1-neg 25.1%

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

    7. fmm-undef 25.1%

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

    8. mul-1-neg 25.1%

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

    9. mul-1-neg 25.1%

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

    10. unsub-neg 25.1%

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

11. Simplified 25.1%

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

12. Taylor expanded in s around 0 25.1%

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

13. Final simplification 25.1%

    \[\leadsto s \cdot \left(\log s - \left(\log \pi + u \cdot -2\right)\right) \]
14. Add Preprocessing

Alternative 7: 25.1% 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 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. associate-*r/ 25.1%

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

   4. associate-*r/ 25.1%

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

   5. div-sub 25.1%

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

   6. distribute-rgt-out-- 25.1%

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

   7. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot \color{blue}{-0.5}}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot -0.5}{s}\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0 25.1%

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

    1. mul-1-neg 25.1%

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

    2. distribute-rgt-neg-in 25.1%

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

    3. mul-1-neg 25.1%

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

    4. distribute-lft-in 25.1%

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

    5. fma-define 25.1%

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

    6. mul-1-neg 25.1%

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

    7. fmm-undef 25.1%

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

    8. mul-1-neg 25.1%

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

    9. mul-1-neg 25.1%

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

    10. unsub-neg 25.1%

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

11. Simplified 25.1%

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

12. Taylor expanded in u around 0 25.1%

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

Alternative 8: 25.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(s * Float32(Float32(-2.0) + Float32(log1p(Float32(Float32(pi) / s)) / u))) * Float32(-u))
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 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. associate-*r/ 25.1%

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

   4. associate-*r/ 25.1%

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

   5. div-sub 25.1%

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

   6. distribute-rgt-out-- 25.1%

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

   7. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot \color{blue}{-0.5}}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot -0.5}{s}\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0 25.1%

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

10. Taylor expanded in u around inf 25.1%

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

    1. +-commutative 25.1%

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

    2. distribute-lft-in 25.1%

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

    3. associate-*r* 25.1%

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

    4. *-commutative 25.1%

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

    5. metadata-eval 25.1%

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

    6. distribute-lft-neg-in 25.1%

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

    7. distribute-lft-neg-in 25.1%

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

    8. *-commutative 25.1%

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

    9. associate-*r* 25.1%

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

    10. mul-1-neg 25.1%

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

    11. distribute-rgt-neg-out 25.1%

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

    12. distribute-neg-in 25.1%

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

    13. distribute-lft-in 25.1%

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

12. Simplified 25.1%

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

13. Final simplification 25.1%

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

Alternative 9: 25.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(u * Float32(Float32(-Float32(-2.0)) - Float32(log1p(Float32(Float32(pi) / s)) / u))))
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 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. associate-*r/ 25.1%

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

   4. associate-*r/ 25.1%

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

   5. div-sub 25.1%

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

   6. distribute-rgt-out-- 25.1%

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

   7. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot \color{blue}{-0.5}}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot -0.5}{s}\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0 25.1%

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

10. Taylor expanded in u around inf 25.1%

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

    1. sub-neg 25.1%

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

    2. log1p-define 25.1%

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

    3. metadata-eval 25.1%

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

12. Simplified 25.1%

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

13. Final simplification 25.1%

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

Alternative 10: 25.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(s * Float32(Float32(-log1p(Float32(Float32(pi) / s))) - Float32(u * Float32(-2.0))))
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. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. associate-*r/ 25.1%

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

   4. associate-*r/ 25.1%

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

   5. div-sub 25.1%

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

   6. distribute-rgt-out-- 25.1%

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

   7. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot \color{blue}{-0.5}}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot -0.5}{s}\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around 0 25.1%

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

10. Final simplification 25.1%

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

Alternative 11: 24.9% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

8. Simplified 25.1%

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

9. Final simplification 25.1%

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

Alternative 12: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(pi) * Float32(Float32(-1.0) + Float32(u * Float32(2.0))))
end

MATLAB

function tmp = code(u, s)
	tmp = single(pi) * (single(-1.0) + (u * single(2.0)));
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. Taylor expanded in s around -inf 24.9%

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

6. Taylor expanded in u around 0 25.1%

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

   1. log1p-define 25.1%

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

   2. associate-*r/ 25.1%

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

   3. associate-*r/ 25.1%

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

   4. associate-*r/ 25.1%

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

   5. div-sub 25.1%

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

   6. distribute-rgt-out-- 25.1%

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

   7. metadata-eval 25.1%

      \[\leadsto \left(-s\right) \cdot \left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot \color{blue}{-0.5}}{s}\right)}{1 + \frac{\pi}{s}}\right) \]

8. Simplified 25.1%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(\frac{\pi}{s}\right) + \frac{4 \cdot \left(u \cdot \frac{\pi \cdot -0.5}{s}\right)}{1 + \frac{\pi}{s}}\right)} \]

9. Taylor expanded in s around inf 10.5%

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

    1. distribute-lft-in 10.5%

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

    2. *-commutative 10.5%

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

    3. associate-*r* 10.5%

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

    4. metadata-eval 10.5%

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

    5. *-commutative 10.5%

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

    6. associate-*r* 10.5%

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

    7. distribute-rgt-out 10.5%

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

    8. *-commutative 10.5%

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

11. Simplified 10.5%

    \[\leadsto \color{blue}{\pi \cdot \left(-1 + u \cdot 2\right)} \]
12. Add Preprocessing

Alternative 13: 12.1% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in s around inf 10.5%

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

   1. cancel-sign-sub-inv 10.5%

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

   2. metadata-eval 10.5%

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

   3. distribute-rgt-out-- 10.5%

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

   4. metadata-eval 10.5%

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

   5. *-commutative 10.5%

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

5. Simplified 10.5%

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

6. Taylor expanded in u around inf 5.1%

   \[\leadsto \color{blue}{2 \cdot \left(u \cdot \pi\right)} \]
7. Step-by-step derivation

   1. associate-*r* 5.1%

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

8. Simplified 5.1%

   \[\leadsto \color{blue}{\left(2 \cdot u\right) \cdot \pi} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 5.1%

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

   2. sqrt-prod 5.1%

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

   3. sqr-neg 5.1%

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

   4. pow2 5.1%

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

   5. sqrt-pow1 12.0%

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

   6. metadata-eval 12.0%

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

   7. pow1 12.0%

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

   8. neg-sub0 12.0%

      \[\leadsto \left(2 \cdot u\right) \cdot \color{blue}{\left(0 - \pi\right)} \]

10. Applied egg-rr 12.0%

    \[\leadsto \left(2 \cdot u\right) \cdot \color{blue}{\left(0 - \pi\right)} \]

11. Final simplification 12.0%

    \[\leadsto \pi \cdot \left(u \cdot \left(-2\right)\right) \]
12. Add Preprocessing

Alternative 14: 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 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 10.3%

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

   1. mul-1-neg 10.3%

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

7. Simplified 10.3%

   \[\leadsto \color{blue}{-\pi} \]
8. Add Preprocessing

Alternative 15: 4.7% 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 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 10.3%

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

   1. mul-1-neg 10.3%

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

7. Simplified 10.3%

   \[\leadsto \color{blue}{-\pi} \]
8. Step-by-step derivation

   1. neg-sub0 10.3%

      \[\leadsto \color{blue}{0 - \pi} \]

   2. sub-neg 10.3%

      \[\leadsto \color{blue}{0 + \left(-\pi\right)} \]

   3. add-sqr-sqrt -0.0%

      \[\leadsto 0 + \color{blue}{\sqrt{-\pi} \cdot \sqrt{-\pi}} \]

   4. sqrt-unprod 4.7%

      \[\leadsto 0 + \color{blue}{\sqrt{\left(-\pi\right) \cdot \left(-\pi\right)}} \]

   5. sqr-neg 4.7%

      \[\leadsto 0 + \sqrt{\color{blue}{\pi \cdot \pi}} \]

   6. sqrt-unprod 4.7%

      \[\leadsto 0 + \color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}} \]

   7. add-sqr-sqrt 4.7%

      \[\leadsto 0 + \color{blue}{\pi} \]

9. Applied egg-rr 4.7%

   \[\leadsto \color{blue}{0 + \pi} \]
10. Step-by-step derivation

    1. +-lft-identity 4.7%

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

11. Simplified 4.7%

    \[\leadsto \color{blue}{\pi} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024184 
(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))))