Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 99.0% → 99.0%

Time: 20.6s

Alternatives: 12

Speedup: 0.5×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{\frac{\pi}{s}}\\ t_1 := \left(\frac{1}{t\_0} + \frac{u}{1 + e^{\frac{\pi}{-s}}}\right) - \frac{u}{t\_0}\\ s \cdot \log \left(\frac{1 + \frac{1}{t\_1}}{-1 + {t\_1}^{-2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (exp (/ PI s))))
        (t_1 (- (+ (/ 1.0 t_0) (/ u (+ 1.0 (exp (/ PI (- s)))))) (/ u t_0))))
   (* s (log (/ (+ 1.0 (/ 1.0 t_1)) (+ -1.0 (pow t_1 -2.0)))))))

C

float code(float u, float s) {
	float t_0 = 1.0f + expf((((float) M_PI) / s));
	float t_1 = ((1.0f / t_0) + (u / (1.0f + expf((((float) M_PI) / -s))))) - (u / t_0);
	return s * logf(((1.0f + (1.0f / t_1)) / (-1.0f + powf(t_1, -2.0f))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))
	t_1 = Float32(Float32(Float32(Float32(1.0) / t_0) + Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s)))))) - Float32(u / t_0))
	return Float32(s * log(Float32(Float32(Float32(1.0) + Float32(Float32(1.0) / t_1)) / Float32(Float32(-1.0) + (t_1 ^ Float32(-2.0))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + exp((single(pi) / s));
	t_1 = ((single(1.0) / t_0) + (u / (single(1.0) + exp((single(pi) / -s))))) - (u / t_0);
	tmp = s * log(((single(1.0) + (single(1.0) / t_1)) / (single(-1.0) + (t_1 ^ single(-2.0)))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around 0 98.8%

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

   1. flip-- 98.7%

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

7. Applied egg-rr 98.9%

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

   1. clear-num 98.9%

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

   2. log-div 99.0%

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

9. Applied egg-rr 99.0%

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(0 - \log \left(\frac{1 + \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}}{{\left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-2} + -1}\right)\right)} \]
10. Step-by-step derivation

    1. neg-sub0 99.0%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{-s}}\\ t_1 := e^{\frac{\pi}{s}}\\ t_2 := -1 - t\_1\\ t_3 := 1 + t\_1\\ \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{1}{t\_3} + \left(\frac{u}{1 + t\_0} + \frac{u}{t\_2}\right)\right)}^{-2}}{1 + \frac{-1}{\left(\frac{u}{t\_3} + \frac{u}{-1 - t\_0}\right) + \frac{1}{t\_2}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI (- s))))
        (t_1 (exp (/ PI s)))
        (t_2 (- -1.0 t_1))
        (t_3 (+ 1.0 t_1)))
   (*
    (- s)
    (log
     (/
      (+ -1.0 (pow (+ (/ 1.0 t_3) (+ (/ u (+ 1.0 t_0)) (/ u t_2))) -2.0))
      (+ 1.0 (/ -1.0 (+ (+ (/ u t_3) (/ u (- -1.0 t_0))) (/ 1.0 t_2)))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / -s));
	float t_1 = expf((((float) M_PI) / s));
	float t_2 = -1.0f - t_1;
	float t_3 = 1.0f + t_1;
	return -s * logf(((-1.0f + powf(((1.0f / t_3) + ((u / (1.0f + t_0)) + (u / t_2))), -2.0f)) / (1.0f + (-1.0f / (((u / t_3) + (u / (-1.0f - t_0))) + (1.0f / t_2))))));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / Float32(-s)))
	t_1 = exp(Float32(Float32(pi) / s))
	t_2 = Float32(Float32(-1.0) - t_1)
	t_3 = Float32(Float32(1.0) + t_1)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(-1.0) + (Float32(Float32(Float32(1.0) / t_3) + Float32(Float32(u / Float32(Float32(1.0) + t_0)) + Float32(u / t_2))) ^ Float32(-2.0))) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(Float32(u / t_3) + Float32(u / Float32(Float32(-1.0) - t_0))) + Float32(Float32(1.0) / t_2)))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = exp((single(pi) / -s));
	t_1 = exp((single(pi) / s));
	t_2 = single(-1.0) - t_1;
	t_3 = single(1.0) + t_1;
	tmp = -s * log(((single(-1.0) + (((single(1.0) / t_3) + ((u / (single(1.0) + t_0)) + (u / t_2))) ^ single(-2.0))) / (single(1.0) + (single(-1.0) / (((u / t_3) + (u / (single(-1.0) - t_0))) + (single(1.0) / t_2))))));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around 0 98.8%

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

   1. flip-- 98.7%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 3: 99.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 25.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.5%

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

6. Taylor expanded in u around 0 24.7%

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

   1. mul-1-neg 24.7%

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

   2. unsub-neg 24.7%

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

   3. associate-*r/ 24.7%

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

   4. associate-*r* 24.7%

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

   5. distribute-rgt-out-- 24.7%

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

   6. metadata-eval 24.7%

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

   7. log1p-define 24.7%

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

8. Simplified 24.7%

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

9. Taylor expanded in s around 0 24.8%

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

    1. neg-mul-1 24.8%

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

    2. *-commutative 24.8%

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

    3. unsub-neg 24.8%

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

11. Simplified 24.8%

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

12. Final simplification 24.8%

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

Alternative 5: 25.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{\pi}{s}\\ 2 \cdot \frac{\pi \cdot u}{t\_0} - s \cdot \log t\_0 \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ PI s)))) (- (* 2.0 (/ (* PI u) t_0)) (* s (log t_0)))))

C

float code(float u, float s) {
	float t_0 = 1.0f + (((float) M_PI) / s);
	return (2.0f * ((((float) M_PI) * u) / t_0)) - (s * logf(t_0));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + Float32(Float32(pi) / s))
	return Float32(Float32(Float32(2.0) * Float32(Float32(Float32(pi) * u) / t_0)) - Float32(s * log(t_0)))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + (single(pi) / s);
	tmp = (single(2.0) * ((single(pi) * u) / t_0)) - (s * log(t_0));
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.5%

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

6. Taylor expanded in u around 0 24.7%

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

   1. mul-1-neg 24.7%

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

   2. unsub-neg 24.7%

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

   3. associate-*r/ 24.7%

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

   4. associate-*r* 24.7%

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

   5. distribute-rgt-out-- 24.7%

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

   6. metadata-eval 24.7%

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

   7. log1p-define 24.7%

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

8. Simplified 24.7%

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

9. Taylor expanded in u around 0 24.7%

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

10. Final simplification 24.7%

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

Alternative 6: 25.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(u * Float32(Float32(Float32(Float32(pi) * Float32(2.0)) / Float32(Float32(1.0) + Float32(Float32(pi) / s))) - Float32(s * Float32(log1p(Float32(Float32(pi) / s)) / u))))
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.5%

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

6. Taylor expanded in u around 0 24.7%

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

   1. mul-1-neg 24.7%

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

   2. unsub-neg 24.7%

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

   3. associate-*r/ 24.7%

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

   4. associate-*r* 24.7%

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

   5. distribute-rgt-out-- 24.7%

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

   6. metadata-eval 24.7%

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

   7. log1p-define 24.7%

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

8. Simplified 24.7%

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

9. Taylor expanded in u around inf 24.7%

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

    1. +-commutative 24.7%

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

    2. mul-1-neg 24.7%

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

    3. unsub-neg 24.7%

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

    4. associate-*r/ 24.7%

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

    5. log1p-define 24.7%

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

    6. associate-/l* 24.7%

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

11. Simplified 24.7%

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

12. Final simplification 24.7%

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

Alternative 7: 25.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u, s)
	return Float32(Float32(Float32(2.0) * Float32(s * u)) - Float32(s * log1p(Float32(Float32(1.0) / Float32(s / Float32(pi))))))
end

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.5%

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

6. Taylor expanded in u around 0 24.7%

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

   1. mul-1-neg 24.7%

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

   2. unsub-neg 24.7%

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

   3. associate-*r/ 24.7%

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

   4. associate-*r* 24.7%

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

   5. distribute-rgt-out-- 24.7%

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

   6. metadata-eval 24.7%

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

   7. log1p-define 24.7%

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

8. Simplified 24.7%

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

   1. clear-num 24.7%

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

   2. inv-pow 24.7%

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

10. Applied egg-rr 24.7%

    \[\leadsto \frac{-4 \cdot \left(\left(s \cdot u\right) \cdot \left(\frac{\pi}{s} \cdot -0.5\right)\right)}{1 + \frac{\pi}{s}} - s \cdot \mathsf{log1p}\left(\color{blue}{{\left(\frac{s}{\pi}\right)}^{-1}}\right) \]
11. Step-by-step derivation

    1. unpow-1 24.7%

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

12. Simplified 24.7%

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

13. Taylor expanded in s around 0 24.7%

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

Alternative 8: 25.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.5%

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

6. Taylor expanded in u around 0 24.7%

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

   1. mul-1-neg 24.7%

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

   2. unsub-neg 24.7%

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

   3. associate-*r/ 24.7%

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

   4. associate-*r* 24.7%

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

   5. distribute-rgt-out-- 24.7%

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

   6. metadata-eval 24.7%

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

   7. log1p-define 24.7%

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

8. Simplified 24.7%

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

9. Taylor expanded in s around 0 24.7%

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

Alternative 9: 25.1% accurate, 4.0× speedup

Math

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

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(Float32(-s) * 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.5%

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

6. Taylor expanded in u around 0 24.7%

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

7. Final simplification 24.7%

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

Alternative 10: 25.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around -inf 24.5%

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

6. Taylor expanded in u around 0 24.7%

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

   1. associate-*r* 24.7%

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

   2. neg-mul-1 24.7%

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

   3. log1p-define 24.7%

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

8. Simplified 24.7%

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

9. Final simplification 24.7%

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

Alternative 11: 11.6% accurate, 61.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in s around inf 11.9%

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

   1. associate--r+ 11.9%

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

   2. cancel-sign-sub-inv 11.9%

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

   3. distribute-rgt-out-- 11.9%

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

   4. *-commutative 11.9%

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

   5. metadata-eval 11.9%

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

   6. metadata-eval 11.9%

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

   7. *-commutative 11.9%

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

7. Simplified 11.9%

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

8. Taylor expanded in u around 0 12.0%

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

   1. neg-mul-1 12.0%

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

   2. +-commutative 12.0%

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

   3. unsub-neg 12.0%

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

   4. associate-*r* 12.0%

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

10. Simplified 12.0%

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

11. Final simplification 12.0%

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

Alternative 12: 11.4% accurate, 216.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in u around 0 11.6%

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

   1. neg-mul-1 11.6%

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

7. Simplified 11.6%

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

Reproduce

herbie shell --seed 2024170 
(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))))