Sample trimmed logistic on [-pi, pi]

Percentage Accurate: 98.9% → 98.9%

Time: 39.6s

Alternatives: 14

Speedup: 0.4×

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{\frac{\pi}{s}}\\ t_1 := \frac{u}{1 + e^{\frac{\pi}{-s}}}\\ t_2 := t\_1 + \frac{1 - u}{t\_0}\\ \left(-s\right) \cdot \log \left(\frac{-1 + {t\_2}^{-3}}{\left(1 + {t\_2}^{-2}\right) + \frac{-1}{\frac{-1 + u}{t\_0} - t\_1}}\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (exp (/ PI s))))
        (t_1 (/ u (+ 1.0 (exp (/ PI (- s))))))
        (t_2 (+ t_1 (/ (- 1.0 u) t_0))))
   (*
    (- s)
    (log
     (/
      (+ -1.0 (pow t_2 -3.0))
      (+ (+ 1.0 (pow t_2 -2.0)) (/ -1.0 (- (/ (+ -1.0 u) t_0) t_1))))))))

C

float code(float u, float s) {
	float t_0 = 1.0f + expf((((float) M_PI) / s));
	float t_1 = u / (1.0f + expf((((float) M_PI) / -s)));
	float t_2 = t_1 + ((1.0f - u) / t_0);
	return -s * logf(((-1.0f + powf(t_2, -3.0f)) / ((1.0f + powf(t_2, -2.0f)) + (-1.0f / (((-1.0f + u) / t_0) - t_1)))));
}

Julia

function code(u, s)
	t_0 = Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))
	t_1 = Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s)))))
	t_2 = Float32(t_1 + Float32(Float32(Float32(1.0) - u) / t_0))
	return Float32(Float32(-s) * log(Float32(Float32(Float32(-1.0) + (t_2 ^ Float32(-3.0))) / Float32(Float32(Float32(1.0) + (t_2 ^ Float32(-2.0))) + Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(-1.0) + u) / t_0) - t_1))))))
end

MATLAB

function tmp = code(u, s)
	t_0 = single(1.0) + exp((single(pi) / s));
	t_1 = u / (single(1.0) + exp((single(pi) / -s)));
	t_2 = t_1 + ((single(1.0) - u) / t_0);
	tmp = -s * log(((single(-1.0) + (t_2 ^ single(-3.0))) / ((single(1.0) + (t_2 ^ single(-2.0))) + (single(-1.0) / (((single(-1.0) + u) / t_0) - t_1)))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

   1. div-inv 99.0%

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

6. Applied egg-rr 99.0%

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

   1. flip3-+ 98.9%

      \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\pi \cdot \frac{1}{s}}}}\right)}^{3} + {-1}^{3}}{\frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\pi \cdot \frac{1}{s}}}} \cdot \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\pi \cdot \frac{1}{s}}}} + \left(-1 \cdot -1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\pi \cdot \frac{1}{s}}}} \cdot -1\right)}\right)} \]

8. Applied egg-rr 99.0%

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3} + -1}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} \cdot -1\right)}\right)} \]
9. Step-by-step derivation

   1. +-commutative 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{\color{blue}{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}}{{\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + \left(1 - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} \cdot -1\right)}\right) \]

   2. associate-+r- 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{\color{blue}{\left({\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + 1\right) - \frac{1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} \cdot -1}}\right) \]

   3. associate-*l/ 99.0%

      \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{\left({\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + 1\right) - \color{blue}{\frac{1 \cdot -1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}}\right) \]

10. Simplified 99.0%

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{\left({\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2} + 1\right) - \frac{-1}{\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right)} \]

11. Final simplification 99.0%

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{-1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-3}}{\left(1 + {\left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{-2}\right) + \frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{\pi}{-s}}}}}\right) \]
12. Add Preprocessing

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{s}}\\ s \cdot \left(-\log \left(-1 + \sqrt[3]{{\left(\frac{1}{1 + t\_0} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{u}{-1 - t\_0}\right)\right)}^{-3}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (let* ((t_0 (exp (/ PI s))))
   (*
    s
    (-
     (log
      (+
       -1.0
       (cbrt
        (pow
         (+
          (/ 1.0 (+ 1.0 t_0))
          (+ (/ u (+ 1.0 (exp (/ PI (- s))))) (/ u (- -1.0 t_0))))
         -3.0))))))))

C

float code(float u, float s) {
	float t_0 = expf((((float) M_PI) / s));
	return s * -logf((-1.0f + cbrtf(powf(((1.0f / (1.0f + t_0)) + ((u / (1.0f + expf((((float) M_PI) / -s)))) + (u / (-1.0f - t_0)))), -3.0f))));
}

Julia

function code(u, s)
	t_0 = exp(Float32(Float32(pi) / s))
	return Float32(s * Float32(-log(Float32(Float32(-1.0) + cbrt((Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + t_0)) + Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))) + Float32(u / Float32(Float32(-1.0) - t_0)))) ^ Float32(-3.0)))))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 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 0 99.0%

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

   1. associate-*r* 99.0%

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

   2. mul-1-neg 99.0%

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

   3. sub-neg 99.0%

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

7. Simplified 99.0%

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

   1. add-cbrt-cube 98.9%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\sqrt[3]{\left(\frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)} \cdot \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}\right) \cdot \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}}} + -1\right) \]

   2. pow1/3 98.5%

      \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{{\left(\left(\frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)} \cdot \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}\right) \cdot \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}\right)}^{0.3333333333333333}} + -1\right) \]

9. Applied egg-rr 98.4%

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{{\left({\left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3}\right)}^{0.3333333333333333}} + -1\right) \]
10. Step-by-step derivation

    1. unpow1/3 99.0%

       \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\sqrt[3]{{\left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3}}} + -1\right) \]

11. Simplified 99.0%

    \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\sqrt[3]{{\left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)\right)}^{-3}}} + -1\right) \]

12. Final simplification 99.0%

    \[\leadsto s \cdot \left(-\log \left(-1 + \sqrt[3]{{\left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \left(\frac{u}{1 + e^{\frac{\pi}{-s}}} + \frac{u}{-1 - e^{\frac{\pi}{s}}}\right)\right)}^{-3}}\right)\right) \]
13. Add Preprocessing

Alternative 3: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

   1. div-inv 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 4: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float u, float s) {
	return -s * logf((-1.0f + (-1.0f / (((-1.0f + u) / (1.0f + expf((((float) M_PI) / s)))) - (u / (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(Float32(Float32(-1.0) + u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) - Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(pi) / Float32(-s))))))))))
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 5: 37.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

   1. div-inv 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in s around inf 37.7%

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

8. Taylor expanded in u around 0 37.7%

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

9. Final simplification 37.7%

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

Alternative 6: 36.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

   1. div-inv 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in s around inf 37.7%

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

8. Taylor expanded in s around inf 36.1%

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

   1. +-commutative 36.1%

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

10. Simplified 36.1%

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

11. Final simplification 36.1%

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

Alternative 7: 24.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\log \left(1 + \frac{4 \cdot \left(u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25\right)}{s}\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (* s (- (log (+ 1.0 (/ (* 4.0 (+ (* u (* PI -0.5)) (* PI 0.25))) s))))))

C

float code(float u, float s) {
	return s * -logf((1.0f + ((4.0f * ((u * (((float) M_PI) * -0.5f)) + (((float) M_PI) * 0.25f))) / s)));
}

Julia

function code(u, s)
	return Float32(s * Float32(-log(Float32(Float32(1.0) + Float32(Float32(Float32(4.0) * Float32(Float32(u * Float32(Float32(pi) * Float32(-0.5))) + Float32(Float32(pi) * Float32(0.25)))) / s)))))
end

MATLAB

function tmp = code(u, s)
	tmp = s * -log((single(1.0) + ((single(4.0) * ((u * (single(pi) * single(-0.5))) + (single(pi) * single(0.25)))) / s)));
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around -inf 24.9%

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

   1. associate-*r/ 24.9%

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

   2. cancel-sign-sub-inv 24.9%

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

   3. distribute-rgt-out-- 24.9%

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

   4. metadata-eval 24.9%

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

   5. metadata-eval 24.9%

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

   6. *-commutative 24.9%

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

5. Simplified 24.9%

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

6. Final simplification 24.9%

   \[\leadsto s \cdot \left(-\log \left(1 + \frac{4 \cdot \left(u \cdot \left(\pi \cdot -0.5\right) + \pi \cdot 0.25\right)}{s}\right)\right) \]
7. Add Preprocessing

Alternative 8: 11.6% accurate, 21.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 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 0 99.0%

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

   1. associate-*r* 99.0%

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

   2. mul-1-neg 99.0%

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

   3. sub-neg 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in s around -inf 11.7%

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

10. Simplified 11.7%

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

11. Final simplification 11.7%

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

Alternative 9: 11.6% accurate, 39.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 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 0 99.0%

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

   1. associate-*r* 99.0%

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

   2. mul-1-neg 99.0%

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

   3. sub-neg 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in s around inf 11.7%

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

   1. metadata-eval 11.7%

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

   2. distribute-lft-neg-in 11.7%

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

   3. associate-*r/ 11.7%

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

   4. distribute-neg-frac2 11.7%

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

10. Simplified 11.7%

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

11. Final simplification 11.7%

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

Alternative 10: 11.6% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf 11.7%

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

   1. cancel-sign-sub-inv 11.7%

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

   2. distribute-rgt-out-- 11.7%

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

   3. metadata-eval 11.7%

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

   4. metadata-eval 11.7%

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

   5. *-commutative 11.7%

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

5. Simplified 11.7%

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

6. Taylor expanded in u around inf 11.7%

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

   1. +-commutative 11.7%

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

   2. mul-1-neg 11.7%

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

   3. unsub-neg 11.7%

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

   4. *-commutative 11.7%

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

8. Simplified 11.7%

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

Alternative 11: 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 99.0%

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

3. Simplified 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 0 99.0%

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

   1. associate-*r* 99.0%

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

   2. mul-1-neg 99.0%

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

   3. sub-neg 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in s around inf 11.7%

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

   1. associate--r+ 11.7%

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

   2. cancel-sign-sub-inv 11.7%

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

   3. *-commutative 11.7%

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

   4. *-commutative 11.7%

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

   5. *-commutative 11.7%

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

   6. *-commutative 11.7%

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

   7. distribute-lft-out-- 11.7%

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

   8. metadata-eval 11.7%

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

   9. metadata-eval 11.7%

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

   10. *-commutative 11.7%

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

   11. distribute-rgt-out 11.7%

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

   12. +-commutative 11.7%

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

   13. associate-*l* 11.7%

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

   14. metadata-eval 11.7%

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

   15. *-commutative 11.7%

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

   16. associate-*l* 11.7%

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

10. Simplified 11.7%

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

11. Final simplification 11.7%

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

Alternative 12: 11.4% accurate, 72.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.4%

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

6. Final simplification 11.4%

   \[\leadsto s \cdot \frac{\pi}{-s} \]
7. Add Preprocessing

Alternative 13: 11.4% accurate, 216.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

5. Taylor expanded in u around 0 11.4%

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

   1. mul-1-neg 11.4%

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

7. Simplified 11.4%

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

Alternative 14: 10.3% accurate, 433.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (u s) :precision binary32 0.0)

C

float code(float u, float s) {
	return 0.0f;
}

Fortran

real(4) function code(u, s)
    real(4), intent (in) :: u
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

function code(u, s)
	return Float32(0.0)
end

MATLAB

function tmp = code(u, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in s around inf 10.3%

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

4. Taylor expanded in s around 0 10.3%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(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))))